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The density matrix theory, the ancestor of density functional theory, provides the immediate framework for Path Integral (PI) development, allowing the canonical density be extended for the many-electronic systems through the density functional closure relationship. Yet, the use of path integral formalism for electronic density prescription presents several advantages: assures the inner quantum mechanical description of the system by parameterized paths; averages the quantum fluctuations; behaves as the propagator for time-space evolution of quantum information; resembles Schrödinger equation; allows quantum statistical description of the system through partition function computing. In this framework, four levels of path integral formalism were presented: the Feynman quantum mechanical, the semiclassical, the Feynman-Kleinert effective classical, and the Fokker-Planck non-equilibrium ones. In each case the density matrix or/and the canonical density were rigorously defined and presented. The practical specializations for quantum free and harmonic motions, for statistical high and low temperature limits, the smearing justification for the Bohr’s quantum stability postulate with the paradigmatic Hydrogen atomic excursion, along the quantum chemical calculation of semiclassical electronegativity and hardness, of chemical action and Mulliken electronegativity, as well as by the Markovian generalizations of Becke-Edgecombe electronic focalization functions – all advocate for the reliability of assuming PI formalism of quantum mechanics as a versatile one, suited for analytically and/or computationally modeling of a variety of fundamental physical and chemical reactivity concepts characterizing the (density driving) many-electronic systems.
density matrix and functionalsFeynman integralpartition functionelectronegativitychemical action and hardnessFokker-Planck equationelectronic localizationIntroduction
In modern conceptual and computational chemistry the Density Functional Theory (DFT) [1–12] plays the central role since its capabilities in providing both structural and reactivity information about the atoms and of their interaction in molecules and nanostructures [13–20]. Yet, the main vehicle stays the electronic density, an observable quantity, which intimately relates with the more abstract quantum mechanical concept of wave function through the basic relationship [3]
ρ(x1)=N∫|Ψ({ri}i=1,N¯)|2ds1dr2…drNwritten for a collection of N-many-electronic system with space-spin coordinates dri = dxidsi, in terms of space and spin coordinates, {xi}i=
1,N and {si}i=
1,N, respectively, in constructing the basic functional (integral) for the charge conservation
N[ρ]=∫ρ(x1)dx1Besides the sub-script index, it is worth noting that with Equations (1) and (2) either the ground state or the valence state(s) of a many body system may be computed by applying the variational principle upon the total or valence density functional energy [7]
E[ρ]=FHK[ρ]+CA[ρ]which in terms of the so called Hohenberg-Kohn density functional [1]
FHK[ρ]=T[ρ]+Vee[ρ]that sums the electronic kinetic T[ρ] and electronic repulsion Vee[ρ], and of the so called chemical action term [7–10]
CA[ρ]=∫ρ(x)V(x)dxaccounts in principle for all observable effects an electronic structure may manifest as the chemical reactivity.
Yet, although accessible experimentally [21], for a deeper comprehension of the physical-chemical phenomenology of bonding the electronic density should be “visualized” also within an analytical framework. In this regard the already consecrated attempt of density matrix made history over a half century by employing the so called ensemble (statistical) density operator [22–26]
ρ^=∑kwk|ϕk〉〈ϕk|which by means of its first order reduced one ρ̂(1) (x1; x1) provides by integration the total number of electrons in the same way as the observable density does [27–31]
∫ρ(1)(x1;x1)dx1=Nthat is formally equivalent with the “trace” operation on the bilocal density matrix ρ̂(1) (x′1; x1)
Tr[ρ^(1)(x′1;x1)]=NFrom this perspective it is clear that having in hand a reliable method to express, for various applied potentials V (x) the bilocal density matrix, from here on called simple as density matrix, yields in fact the electronic density itself. Fortunately the quantum mechanically formalism had advanced a rigorous way of expressing the density matrix since recognizing it as a specialization of the more general time evolution quantum amplitude, a.k.a. the quantum propagator or the Green function [32–36]
(xbtb;xata)=〈xb|exp(−βH^)|xa〉
Moreover, within the thermal-temporal (quantum mechanical-to quantum statistical) Wick transformation
tb−ta=−iℏβ=−iℏkBTwith kB the Boltzmann constant, T the sample temperature, and ħ - the Planck constant, there is immediate to introduce the partition function of a system as the analytically time continued integral
Z=Tr[exp(−βH^)]=∫−∞+∞dx〈x|exp(−βH^)|x〉=∫−∞+∞dx(xtb;xta)to release the uni-particle density
ρ(x)=NZ〈x|exp(−βH^)|x〉while fulfilling the normalization constraint given by Equation (2).
The competition between the variation of density functional of energy (3) and the density itself (12), i.e., the ratio between the global and local influences on a many-electronic system (equilibrium, aromaticity, etc.) is quantified by the modern electronegativity index and by its chemical hardness companion [3,5–12], given respectively as
χ=−(δE[ρ]δρ)V(x)=ct.η=−(δχ[ρ]δρ)V(x)=ct.which may assume various analytical realizations within density functional theory [7,9], while playing the crucial roles in driving chemical bonding and reactivity [12,20]; this may be also immediately seen from Equations (13) when electronegativity is further identified with the negative of the chemical potential of a given system (χ =−μ) [6]: if electronegativity is assimilated with the chemical potential, the chemical hardness – as its derivative – corresponds with the chemical force; thus, they together constitute the minimal necessary set of indices to consistently model the complete scenario of chemical reactivity, from encountering adducts to the stabilized products [9]. For these reasons they will be systematically presented in this review as application for different levels of path integral approximations in either matrix density or density computations.
However, it is worth comment that the above factorization only apparently assumes the many-particle system without internal interaction (exchange and correlation), while all these effects are to be incorporated in the way the bare applied potential is replaced with an effective one or by performing a variational (upon the) perturbation procedure for optimizing the (bilocal) density matrix
ρ(1)(xb;xa)=∫xaxbDx(τ)exp{−1ℏ∫0βdτH^(x(τ),x˙(τ),τ)}=∫xaxbDx(τ)exp{−1ℏ∫0βdτ[mx˙2(τ)2+V(x(τ))]}in accordance with above recipe. Equation (14) is nevertheless nothing else than the so called path integral of the density matrix or of the time evolution quantum amplitude, while ∫ Dx(τ) is a complex symbol of integration over (parameterized, quantum) paths that when unfolded recovers the normal shape of integration, i.e., with separated variables of integration, e.g., Equations (117) or (316) and Refs. [37–39]. The way in which this path integral is computed for various working potentials will generate the analytical solution of the quantum amplitude, and implicitly to the density matrix, from where the electronic density is immediately found by the identity ρ(xa) = ρ(1) (xb = xa; xa). Then, having the electronic density any known or approximated density functional may be evaluated and employed in describing the chemical structure and reactivity in an analytical manner, while allowing better conceptual understanding of the obtained models and predictions. With these, the motivation behind the present project becomes clear: computing the electronic density relays in fact on evaluating the associate path integral for a given potential; moreover, the many-body effects are to be resumed in rewriting the applied potential into an effective one (one way) or to advance the so called variational perturbation methodology in order the inter-particle effects be accommodated [39–48], being this latter approach left for further communications.
Consequently, the review unfolds on bigger scale the ideas here presented: it starts with the basic properties of the density matrix and showing how the path integral concept arises naturally in this framework. Then, a more formal introduction of path integral methodology is presented in the spirit of Richard Feynman, its main promoter; and the use of path integrals is exemplified in computing semi-classical time evolution amplitudes with application on atomic electronegativity and chemical hardness reactivity indices. The simplified many-body approach is then given through exposing the Feynman-Kleinert algorithm for effective potentials, with application on computing atomic Mulliken electronegativities, while the non-equilibrium Fokker-Planck approach is exposed and applied in the context of Markovian stochastic motion within the anharmonic potential and then extended to modeling the electronic localization through computing several Markovian electronic functions while comparing them with the circulating Becke-Edgecombe one [49,50]. This way, a fruitful step is hopefully made towards unifying the physical-chemical principles of electronic structure and reactivity on a meaningful quantum basis.
From Density Matrix to Path IntegralOn Mono-, Many-, and Reduced- Electronic Density Matrices
Given a spectral representation {|n〉}n∈N for a set of quantum mono-electronic states
|ϕk〉=∑nckn|n〉one may employ its closure relation:
1^=∑n|n〉〈n|to generally express the average of an observable (i.e., the operator Â) on a selected state as
〈A^〉k=〈ϕk|A^|ϕk〉〈ϕk|ϕk〉=∑n,n′〈ϕk|n′〉〈n′|A^|n〉〈n|ϕk〉∑n〈ϕk|n〉〈n|ϕk〉=∑n,n′cknckn′*〈n′|A^|n〉∑n|ckn|2while for the observable average over the entire sample the individual weight wk should be counted to provide the statistical result
〈A^〉=∑kwk〈A^〉k∑kwk
Similarly, when rewriting the global average as
〈A^〉=∑n,n′〈n|ρ^|n′〉〈n′|A^|n〉∑kwkwe recognize the density matrix elements〈n|ρ^|n′〉=∑kwkcknckn′*∑n|ckn|2which provides the density operatorρ^=∑n,n′|n〉〈n|ρ^|n′〉〈n′|=∑kwk∑n|ckn|2(∑nckn|n〉)(∑n′〈n′|ckn′*)=∑kwk∑n|ckn|2|ϕk〉〈ϕk|with the sum of diagonal matrix elements (yielding the “trace” function)
Trρ^=∑n〈n|ρ^|n〉=∑kwk∑n|ckn|2∑n|ckn|2=∑kwkwhile the searched operatorial average now becomes
〈A^〉=∑n,n′〈n|ρ^|n′〉〈n′|A^|n〉∑kwk=∑n〈n|ρ^A^|n〉∑n〈n|ρ^|n〉=Tr(ρ^A^)Trρ^
Note that through the above deductions the double (independent) averages technique was adopted exploiting therefore the associate sums inter-conversions to produce the simplified results [51–53]. Yet, this technique is equivalent with quantum mechanically factorization of the entire Hilbert space into sub-spaces or, at the limit, into the subspace of interest (that selected to be measured, for instance) and the rest of the space, being thus this approach equivalent with a system-bath sample; this is useful for better understanding the stochastic phenomena – to be latter exposed – that underlay to open quantum systems. Therefore, such mechanism may be considered as belonging to the physical foundations for the chemical reactivity.
Next, in the case the concerned quantum states are eigen-states, they fulfill the normalization constraint
δkk′=〈ϕk|ϕk′〉=∑nckn*ck′n⇒∑n|ckn|2=1on which base the above density operator now reads with the same form as presented din Introduction, Equation (6), from where there appears that the eigen-equation for it looks like
ρ^|ϕk〉=∑k′wk′|ϕk′〉〈ϕk′|ϕk〉︸δk′k=wk|ϕk〉giving with the eigen-values (as the diagonal elements) as
〈ϕk|ρ^|ϕk〉=wkas the observed values of the averaged density operator. Since they are weights of probability they have to naturally fulfill the closure probability relationship over the entire sample
∑kwk=1from where follows the “normalization of density operator” through its above Trace property of Equation (22)Trρ^=1
Moreover, in these eigen-conditions, the operatorial average further reads from Equation (23)〈A^〉=Tr(ρ^A^)
Now, there appears with better clarity the major role the density operator plays in quantum measurements, since it convolutes with a given operator to produce its (averaged) measured value on the prepared eigen-states. Nevertheless, when the so called pure states are employed or prepared, the preceding distinction between the subsystem and system vanishes, and the density operator takes the pure quantum mechanical form of an elementary projector
ρ^=|ϕ〉〈ϕ|≡Λ^
This is a very useful expression for considering it associated with the mono-density operators when many-fermionic systems are treated, although a similar procedure applies for mixed (sample) states as well. It is immediate to see that for N formally independent partitions the Hilbert space corresponding to the N-mono-particle densities on pure states, we have through Equations (22), (27), (28) and (30)Λ^i=|ϕi〉〈ϕi|,TrΛ^i=1,i=1,N¯producing the total operator – projector constructed by the sum
Λ^N=∑i=1NΛ^iwhile correctly normalized to the total number of particles
TrΛ^N=Tr(∑i=1NΛ^i)=∑i=1NTrΛ^i︸1=N
Yet, the anti-symmetric restriction the N-fermionic state may be accounted from the mono-electronic states through considering Slater permutated (Pα) products [8,35]
|ΦN〉=1N!∑Pα(−1)PαPα[∏i=1N|ϕi〉]for constructing the N-electronic density operator
ρ^(N)=|ΦN〉〈ΦN|with which help the N × N density matrix writes as (in coordinate representation)
ρ(N)(x′1…x′N;x1…xN)=〈x′1…x′N|ρ^(N)|x1…xN〉=〈x′1…x′N|ΦN〉〈ΦN|x1…xN〉=1N!∑P′α(−1)P′αP′α[∏i=1N〈x′i|ϕi〉]∑Pα(−1)PαPα[∏i=1N〈ϕi|xi〉]=1N!∑P′α(−1)P′αP′α[∏i=1Nϕi(x′i)]∑Pα(−1)PaPα[∏i=1Nϕi*(xi)]
However, in practice, due to the fact the multi-particle operators associate with number of systemic properties less than the total number of particle, say of order p < N, worth working with the p-order reduced density matrix introduced as
ρ(p)(x′1…x′p;x1…xp)=(Np)∫ΦN*(x1…xN)ΦN(x′1…x′N)∏j=p+1Ndxjwith the following features [27].
○ First order Löwdin reduction
ρ(p)(x′1…x′p;x1…xp)=1p!det[ρ(1)(x′k;xk)]=1p!|ρ(1)(x′1;x1)ρ(1)(x′1;x2)⋯ρ(1)(x′1;xp)ρ(1)(x′2;x1)ρ(1)(x′2;x2)⋯ρ(1)(x′2;xp)⋮⋮⋮ρ(1)(x′p;x1)ρ(1)(x′p;x2)⋯ρ(1)(x′p;xp)|
where the first order density matrix casts as abstracted from general definition
ρ(1)(x′1;x1)=N∫ΦN*(x1…xN)ΦN(x′1…x′N)∏j=2Ndxj
With these concepts it is worth noting the major importance that the first order density plays in computing the higher order reduced density matrices that in turn enter the operatorial averages, for instance
〈A^〉=∑p=1NTr(p)[A^(x1…xp)ρ^(p)]
A special reference may be made in regard of the free-relativist treatment of many-electronic atoms, ions, bi- or poly- atomic molecules governed by the working Hamiltonian
H^=e2∑G<HZGZHRGH+∑i=1Npi22m−e2∑i,GZGriG+e2∑i<j1rijwhose terms are represented the inter-nuclear repulsion (only for molecules), free electronic motion, electron-nuclei Coulombic attraction, and inter-electronic Coulombian repulsion, respectively. For such Hamiltonian the average value is computed through considering electronic density of the first or second order only there where the electronic influence is present, while the degree of matrix density is fixed by the type of electronic interaction
〈H^〉=e2∑G<HZGZHRGH+12m∫p12ρ(1)(x′1;x1)|x′1=x1dx1−e2∑GZG∫ρ(1)(x′1;x1)r1G|x′1=x1dx1+e2∫∫ρ(2)(x′1,x′2;x1,x2)r12|x′1=x1x′2=x2dx1dx2
It is obvious that although the second order reduced matrix has appeared, its general form
ρ(2)(x′1,x′2;x1,x2)=(N2)∫ΦN*(x1…xN)ΦN(x′1…x′N)∏j=3Ndxjmay be further reduced to the first one through the above determinant rule, see Equation (40)ρ(2)(x′1,x′2;x1,x2)=12!|ρ(1)(x′1;x1)ρ(1)(x′1;x2)ρ(1)(x′2;x1)ρ(1)(x′2;x2)|=12[ρ(1)(x′1;x1)ρ(1)(x′2;x2)−ρ(1)(x′1;x2)ρ(1)(x′2;x1)]this way emphasizing on the importance of the first order reduced matrix knowledge.
The astonishing physical meaning behind this formalism relays in the fact that any multi-particle interaction (two-particle interaction included) may be reduced to the single particle behavior; in other terms, vice-versa, the appropriate perturbation (including strong-coupling) of the single particle evolution carries the equivalent information as that characterizing the whole many-body system.
In fact, the power of the density matrix formalism resides in reducing a many-body problem to the single particle density matrix, abstracted from the single Slater determinant of Equation (36) known as the Fock-Dirac matrixρFD(1)(x′1;x1)=∑i=1Nϕi*(x1)ϕi(x′1)that along the associate operator
ρ^FD(1)=∑i=1N|ϕi〉〈ϕi|considerably simplifies the quantum problem to be solved. Let’s illustrate this by firstly quoting that Fock-Dirac density operator of Equation (48) has two fundamental properties, namely:
○ The idempotency
ρ^FD(1)ρ^FD(1)=∑i,j=1N|ϕi〉〈ϕi|ϕj〉︸δij〈ϕj|=∑i=1N|ϕi〉〈ϕi|=ρ^FD(1)
○ The normal additivity, see Equations (33)Trρ^FD(1)=Tr(∑i=1N|ϕi〉〈ϕi|)=Tr(∑i=1NΛ^i)=N
while having the corresponding coordinate integral representations:
Remarkably, the last two identities may serve as the constraints when minimizing the above Hamiltonian average, here appropriately rewritten employing Equations (44) and (46), and where all external applied potential, were resumed under generic V (x1) quantity producing the actual so called Hartree-Fock trial density matrix energy functionalEHF[ρFD(1)]=∫[−ℏ22m∇12+V(x1)]ρFD(1)(x′1;x1)|x′1=x1dx1+e22∫∫1r12[ρFD(1)(x1;x1)ρFD(1)(x2;x2)−ρFD(1)(x1;x2)ρFD(1)(x2;x1)]dx1dx2obeying the (Lagrange) variational principle
δ{EHF[ρFD(1)]−∫∫α(x′1;x1)[∫ρFD(1)(x′1;x″1)ρFD(1)(x″1;x1)dx″1−ρFD(1)(x′1;x1)]dx′1dx1−β[∫∫δ(x′1−x1)ρFD(1)(x′1;x1)dx′1dx1−N]}=0
Performing the functional derivative respecting the Fock-Dirac electron density in (54) one gets the equivalent expression
δEHF[ρFD(1)]δρFD(1)(x′1;x1)−∫ρFD(1)(x′1;x¯)α(x¯;x1)dx¯−∫α(x′1;x¯¯)ρFD(1)(x¯¯;x1)dx¯¯+α(x′1;x1)−βδ(x′1−x1)=0which eventually transcribes at the operatorial level as
F^−ρ^FD(1)α^−α^ρ^FD(1)+α^−β1^δ=0with
1^δ staying for the operator of the delta-Dirac matrix δ(x′1 – x1), with F̂ the Fock corresponding to the coordinate matrix representation [3]
F(x′1;x1)=δEHF[ρFD(1)]δρFD(1)(x′1;x1)=[−ℏ22m∇12+V(x1)]δ(x′1−x1)+δ(x′1−x1)e2∫1r12ρFD(1)(x2;x2)dx2−e2r1′1ρFD(1)(x′1;x1)︸EXCHANGECONTRIBUTION
Giving the idempotency property of Equation (49), through multiplying Equation (56) on its right side with the Fock-Dirac density operator
F^ρ^FD(1)−ρ^FD(1)α^ρ^FD(1)−α^(ρ^FD(1))2︸ρ^FD(1)+α^ρ^FD(1)−β1^δρ^FD(1)=0while doing the same on left side
ρ^FD(1)F^−(ρ^FD(1))2︸ρ^FD(1)α^−ρ^FD(1)α^ρ^FD(1)+ρ^FD(1)α^−βρ^FD(1)1^δ=0and subtracting the results, one gets the equation
F^ρ^FD(1)−ρ^FD(1)F^=0that is equivalently of saying that Fock energy operator commutes with the Fock-Dirac density operator
⌊F^,ρ^FD(1)⌋=0meaning that they both admit the same set of eigen-functions. This is nevertheless the gate for obtaining the density (matrix) functional energy expressions by means of finding the density (matrix) eigen-solutions only.
Yet, condition (59b) is indeed a workable (reduced) condition raised from optimization of the averaged Hamiltonian of a many-electronic system, since the more general one referring to the whole Hamiltonian, known as the Liouville or Neumann equation, is obtained employing the temporal Schrödinger equation:
iℏ∂∂t|φi〉=H^|φi〉to the evolution equation of Fock-Dirac density operator evolution
iℏ∂∂tρ^FD(1)=∑i=1Niℏ∂∂t(|φi〉〈φi|)=∑i=1Niℏ(∂∂t|φi〉)︸H^|φi〉〈φi|+∑i=1N|φi〉iℏ(∂∂t〈φi|)︸〈φi|H^=H^(∑i=1N|φi〉〈φi|)−(∑i=1N|φi〉〈φi|)H^=[H^,ρ^FD(1)]
Lastly, it should be noted that all above properties may be rewritten since considering the mixed p-order reduced matrix with the form
ρ(p)(x′1…x′p;x1…xp)=∑kwkρk(p)(x′1…x′p;x1…xp)as a natural extension of the pure states. However, the sample statistical effects may be better considered by further expressing the electronic density operator and its matrix, the associate equation and the properties for systems in thermodynamic equilibrium (with environment) – a matter addressed in next section.
Canonical Density, Bloch Equation, and the Need of Path Integral
For a quantum system obeying the N-mono-electronic eigen-equations
H^|φk〉=Ek|φk〉the probability of finding one particle in the state |φk〉 at thermodynamical equilibrium with others, while all states are considered as a closed supra-system with no mass or charge transfer allowed, is given by the canonical distribution [36]
wk=1Z(β)exp(−βEk)providing the mixed Fock-Dirac density with the form
ρ^N(β)=∑k=1N1Z(β)exp(−βEk)|φk〉︸exp(−βH^)|φk〉〈φk|=∑k=1N1Z(β)exp(−βH^)|φk〉〈φk|=1Z(β)exp(−βH^)∑k=1N|φk〉〈φk|=1Z(β)exp(−βH^)∑k=1N(∑nckn|n〉)(∑n〈n|ckn*)=1Z(β)exp(−βH^)∑k=1N(∑n|ckn|2)︸1(∑n|n〉〈n|)︸1=NZ(β)exp(−βH^)
This is a very interesting and important result motivating the quantum statistical approach in determining the density of states since it corresponds to the N-sample particles throughout a simple N-multiplication. Note that Equation (64) is in full agreement with that introduced in Equation (12), and very well suited for handling since respecting the DFT custom normalization of Equation (2), while its normalization factor, the partition function Z(β), follows from such constraint with the consecrated expression
Z(β)=Tr[exp(−βH^)]=∫〈x|e−βH^|x〉dx
The recognized importance of partition functions in computing the internal energy as the average of the Hamiltonian of the system:
UN:=〈H^〉=Tr[ρ^N(β)H^]=NZ(β)Tr[H^exp(−βH^)]=−N∂∂βlnZ(β)or to evaluate the free energy of the system
FN=−N1βlnZ(β)is thus transferred to the knowledge of the closed evolution amplitude 〈x|e–βĤ|x〉, that at its turn is based on the genuine (not-normalized) density operatorρ^⊗(β)=exp(−βH^)sometimes called also like canonic density operator.
The great importance of density operator of Equation (68) is immediately visualized in three ways:
○ It identifies the evolution operator
U^(tb,ta)=exp[−iℏH^(tb−ta)]on the ground of Wick equivalence relationship of Equation (10), which allows the transformation of the Schrödinger into Heisenberg or Interaction pictures for appropriately describing the quantum interactions [53];
○ It produces the so called Bloch equation [21] by taking its β derivative
−∂ρ^⊗(β)∂β=H^ρ^⊗(β)that identifies with the Schrödinger equation for genuine density operator
iℏ∂∂tρ^⊗(β)=H^ρ^⊗(β)through the same Wick transformation given by Equation (10), thus providing the quantum-mechanically to quantum-statistical equivalence;
○ Fulfills the (short times, higher temperature) so called Markovian limiting condition
limβ→0ρ^⊗(β)=1^a very useful constraint for developing either the perturbation or the variational formalism respecting electronic density and/or partition function, see below.
In the frame of coordinate representation the Bloch problem, i.e., the differential equation together with the initial (Cauchy) condition, looks like
{−∂∂βρ⊗(x′;x;β)=H^ρ⊗(x′;x;β)limβ→0ρ⊗(x′;x;β)=δ(x′−x)
Solution of this system is a great task in general, unless the perturbation method is undertaken for writing the Hamiltonian as the sum of free and small interaction components
H^=H^0+H^1for which the free Hamiltonian solution is completely known, say
ρ^0(β)=exp(−βH^0)
In these conditions, one may firstly write
∂∂β(eβH^0ρ^⊗)=H^0eβH^0ρ^⊗+eβH^0∂ρ^⊗∂β︸−H^ρ^⊗=H^0eβH^0ρ^⊗−(H^0+H^1)eβH^0ρ^⊗=−eβH^0H^1ρ^⊗where the inter-Hamiltonian components were considered to freely commute as per wish; then, the Equation (75) is integrated on the realm [0, β] to get
eβH^0ρ^⊗(β)−1^=−∫0βeβ′H^0H^1(β′)ρ^⊗(β′)dβ′that may be rearranged under the perturbative fashion
ρ^⊗(β)=ρ^0(β)−∫0βρ^0(β−β′)H^1(β′)ρ^⊗(β′)dβ′in the form reminding by the Lippmann-Schwinger equation for the perturbed dynamical wave-function [32], with ρ̂0 (β − β′) playing the role of the retarded Green function G0 (tb – ta) [34]. Yet, expression (77) may be further generalized for the p-order approximation by choosing various p-paths of spanning the statistical realm [0, β] by intermediate sub-intervals
β=βn+1>βn>…>β2>β1>β0=0thus leaving with the expansion
ρ^⊗(β)=ρ^0(β)+∑l=1n(−1)l∫0βdβl∫0βldβl−1…∫0β2dβ1×ρ^0(β−βl)[H^1(βl)ρ^0(βl−βl−1)]⋯[H^1(β2)ρ^0(β2−β1)]H^1(β1)ρ^0(β1)correspondingly written in coordinate representation
ρ⊗(x′;x;β)=ρ0(x′;x;β)+∑l=1n(−1)l(∫0βdβl∫0βldβl−1…∫0β2dβ1)×(∫−∞+∞⋯∫−∞+∞∏j=1ldxj)×ρ0(x′,xl;β−βl)[H^1(βl)ρ0(xl;xl−1;βl−βl−1)]⋯[H^1(β2)ρ0(x2;x1;β2−β1)]H^1(β1)ρ0(x1;x;β1)once a parallel slicing of the spatial interval [x', x] is considered through the subdivisions
x′=xn+1>xn…>x2>x1>x0=x
Such slicing procedure in solving the Bloch equation (72) for canonic density solution (80) seems an elegant way of avoiding the self-consistent equation (77). Therefore, it may further employed through reconsidering the problem (72) in a slightly modified variant, namely within the temporal approach
{−ℏ∂∂uρ⊗(x′;x;u)=H(x′)ρ⊗(x′;x;u)ρ⊗(x′;x;u=0)=1where the variable u = ħβ was considered for the time dimension.
Now, in the first instance, the new problem (82) has the formal total solution
ρ⊗(x′;x;u)=exp[−1ℏH(x′)u]that being of exponential type allows for direct slicing through factorization. That is, when considering the space partition given by coordinate cuts of (81), and assuming that the times flows equally on each sub-interval in quota of ε, u = (n + 1)ε, the density solution (83) may be written as a product of intermediary solutions towards the path integral representation
ρ⊗(x′;x;u)=∏j=0n+1exp[−1ℏH(xj)ε]=∫⋯∫ρ⊗(x′;xl;ε)ρ⊗(xl;xl−1;ε)⋯ρ⊗(x1;x;ε)∏j=1ndxj→ε→0(n+1)ε=un→∞∫…∫Λ[x(u)]Dx(u)where the chained covariant density product was introduced
Λ[x(u)]=limn→∞ε→0(n+1)ε=uρ⊗(x′;xl;ε)ρ⊗(xl;xl−1;ε)⋯ρ⊗(x1;x;ε),along the extended integration metric
Dx(u)=limn→∞∏j=1ndxj
The general canonic solution (84) is viewed as the path integral solution for the Bloch equation (82), being therefore as a necessity when looking to general solutions for a given Hamiltonian; it gives the general solution for electronic density (68) since accounting for all path connecting two end-points either in space and time (or temperatures) through in principle an infinite intermediary points; this way the resulted path integral comprises all quantum information contained by the particle’ evolution between two states in thermodynamical equilibrium with environment (or the other mono-particle states). However, once having the canonical density evaluated by its path integral, the associate mixed density matrix may be immediately written employing the operatorial form (64) to the actual spatial representation
ρN(x′;x;u)=NZ(u)ρ⊗(x′;x;u)with the path integral based partition function written in accordance with Equation (65)Z(u)∫ρ⊗(x;x;u)dxassuring the preservation of the general DFT normalization condition
∫ρN(x;x;u)dx=N
This way, the general algorithm linking the path integral to the density matrix and to the electronic density, most celebrated DFT quantity in computing various density functionals (energies, reactivity indices) for characterizing chemical structure and reactivity – was established, while emphasizing the basic role the path integral evaluation has towards a conceptual understanding of many-electronic quantum systems in their dynamics and interaction.
Being thus established the role and usefulness of path integral in density functional theory the next section will give more insight in appropriately defining (constructing) path integral such that to further facilitate its practical evaluation for electronic systems of physical-chemical interest.
Feynman’s Path Integral of Evolution AmplitudeConstruction of the General Path Integral
Through reconsidering the slicing of (81) also for the time interval [tb, ta]
tb=tn+1>tn>…>t2>t1>t0=tawith the spatial ending points recalled as x′ = xb, x = xa the quantum propagator of Equation (9), within the Wick equivalence (10)
(xbtb;xata)=〈xb|exp(−iℏ(tb−ta)H^)|xa〉may be firstly rewritten in terms of associate evolution operator
U^(tb−ta)=exp(−iℏ(tb−ta)H^)to successively become
(xbtb;xata)=〈xb|U^(tb−ta)|xa〉=〈xb|U^(tb,tn)U^(tn,tn−1)…U^(tj,tj−1)…U^(t2,t1)U^(t1,ta)|xa〉=∏j=1n[∫−∞+∞dxj]∏j=1n+1(xjtj;xj−1tj−1)where the last form was obtained when n-times the complete eigen-coordinate set
1^=∫−∞+∞|xj〉〈xj|dxj,j=1,n¯was introduced for each pair of events with the elementary propagator between them
(xjtj;xj−1tj−1)=〈xj|exp(−iℏ(tj−tj−1)H^)|xj−1〉=〈xj|exp(−iℏεH^)|xj−1〉on the elementary time interval
ε=tj−tj−1=tb−tan+1>0
Now, the elementary quantum evolution amplitude (95) is to be evaluated, firstly by reconsidering the eigen-coordinate unitary operator, in the working form
1^x=∫−∞+∞|x〉〈x|dxto separate the operatorial Hamiltonian contributions to the kinetic and potential ones
H^=T^+V^yielding:
(xjtj;xj−1tj−1)≅〈xj|e−iℏεV^(x^,tj)1^xe−iℏεT^(p^,tj)|xj−1〉=∫−∞+∞〈xj|e−iℏεV^(x^,tj)|x〉〈x|e−iℏεT^(p^,tj)|xj−1〉dxwhere we have used the first order limitation of the Baker-Hausdorff formula
e−iℏε[V^(x^,tj)+T^(p^,tj)]=e−1ℏεV^(x^,tj)e−iℏεT^(p^,tj)+O(ε2)︸≅0by assuming the second order of elementary time intervals as vanishing
ε2≅0
Next, each obtained working energetic contribution is separately evaluated: for kinetic contribution the insertion of the momentum complete eigen-set
1^pj=∫−∞+∞|pj〉〈pj|dpjyields
〈x|e−iℏεT^(p^,tj)|xj−1〉=∫−∞+∞〈x|e−iℏεT^(p^,tj)|pj〉〈pj|xj−1〉dpj=∫−∞+∞e−iℏεT(pj,tj)〈x|pj〉︸12πℏeiℏpjx〈pj|xj−1〉︸12πℏe−iℏpjxj−1dpj=∫−∞+∞eiℏpj(x−xj−1)e−iℏεT(pj,tj)dpj2πℏwhile for potential elementary amplitude one gets
〈xj|e−iℏεV^(x^,tj)|x〉=e−iℏεV(x,tj)〈xj|x〉︸δ(xj−x)=δ(xj−x)e−iℏεV(x,tj)With relations (103) and (104) back in (100) the elementary propagator takes the form
(xjtj;xj−1tj−1)=∫−∞+∞(∫−∞+∞eiℏpj(x−xj−1)e−iℏεT(pj,tj)dpj2πℏ)(δ(xj−x)e−iℏεV(x,tj))dx=∫−∞+∞dpj2πℏexp[ipj(xj−xj−1)ℏ−iεT(pj,tj)+V(xj,tj)ℏ]=∫−∞+∞dpj2πℏexp{iℏ[pj(xj−xj−1)−εH(xj,pj,tj)]}Replacing the elementary quantum amplitude (105) back into the global one given by Equation (93), it takes the form
(xbtb;xata)≅∏j=1n[∫−∞+∞dxj]∏j=1n+1∫−∞+∞dpj2πℏexp{εiℏ[pjxj−xj−1ε−H(xj,pj,tj)]}=(∏j=1n[∫−∞+∞dxj])(∏j=1n+1[∫−∞+∞dpj2πℏ])exp{iℏε∑j=1n+1[pjxj−xj−1ε−H(xj,pj,tj)]}which for an infinitesimal temporal partition, i.e.n→∞;ε→0behaves like the Feynman path integral of the quantum propagator [37,40,41,54–60]
(xbtb;xata)≡∫x(ta)=xax(tb)=xbDxDpexp{iℏS[x,p,t]}through considering the limiting notations for the path integral measure
limn→∞(∏j=1n[∫−∞+∞dxj])(∏j=1n+1[∫−∞+∞dpj2πℏ])≡∫x(ta)=xax(tb)=xbDxDpand for the involved action
limn→∞ε→0ε∑j=1n+1[pjxj−xj−1ε−H(xj,pj,tj)]≡(96)∫tatb[p(t)x˙(t)−H(x(t),p(t),t)]dt=∫tatbL(x(t),p(t),t)dt=S[x,p,t]Note that the results (108)–(110) give similar quantum information for the quantum evolution of a system as previously found with the mean of density matrix (84)–(86), yet in a more formal and general way throughout accounting all histories (possibilities for linking two events in time-space) for a quantum evolution [61–63]
(xbtb;xata)=∑ALLHISTORIESexp{iℏS[x,p,t]}thus being suitable to be implemented in the N-particle density functional scheme (87)–(89) once it is analytically computed.
For achieving such goal, a more practical form of the Feynman integral may be obtained once the Hamiltonian is implemented as
H=p22m+V(x,t)leaving the action (110) unfolded as
S[x,p,t]=∑j=1n+1[pj(xj−xj−1)−εpj22m−εV(xj,tj)]=∑j=1n+1[−ε2m(pj−xj−xj−1εm)2+εm2(xj−xj−1ε)2−εV(xj,tj)]from where the momentum integrals in (109) is immediately solved to be
∫−∞+∞dpj2πℏexp{−iℏε2m(pj−xj−xj−1εm)2}=m2πℏiεby formally applying the Poisson formula
∫−∞+∞exp(−Ξy2)dy=πΞ.
The remaining quantum evolution amplitude reads as the spatial path integral only
(xbtb;xata)=(∏j=1n[∫−∞+∞dxj])(∏j=1n+1[m2πℏiε])exp{iℏ∑j=1n+1[εm2(xj−xj−1ε)2−εV(xj,tj)]}=12πℏiε/m∏j=1n[∫−∞+∞dxj2πℏiε/m]exp{iℏε∑j=1n+1[m2(xj−xj−1ε)2−V(xj,tj)]}→ε→0n→∞∫x(ta)=xax(tb)=xbD′xexp{iℏS[x,x˙,t]}assuming the actual modified measure of integration
∫x(ta)=xax(tb)=xbD′x≡limn→∞ε→0(12πℏiε/m∏j=1n[∫−∞+∞dxj2πℏiε/m])and the working action
S[x,x˙,t]=∫tatbL(x,x˙,t)dt=∫tatb[mx˙22−V(x,t)]dt
Note that when the partition function (88) is under consideration, other path integral out of (116) has to be introduced by means of closed space coordinates, namely
Z(tb;ta)=∫−∞+∞(xtb;xta)dx=∫x(ta)=x(tb)D″xexp{iℏS[x,x˙,t]}noting the new integration measure
∫x(ta)=x(tb)D″x≡limn→∞(∏j=1n+1[∫−∞+∞dxj2πℏiε/m])
Therefore, at the first instance, some of the main advantages dealing with path integrals relay on following features:
○ Attractive conceptual representation of dynamical quantum processes without operatorial excursion;
○ Allows for quantum fluctuation description in analogy with thermic description, through changing the temporal intervals with the thermodynamical temperature by means of Wick transformation (10), i.e., transforming quantum mechanical (QM) into quantum statistical (QS) propagators
(xbtb;xata)QM=∫x(ta)=xax(tb)=xbD′x(t)exp{iℏ∫tatb[mx˙22−V(x(t),t)]dt}↓|t:=−iτ,τ=ℏβdt=−idτddt=iddτ(xbℏβ;xa0)QS=∫x(0)=xax(ℏβ)=xbD′x(τ)exp{−1ℏ∫0ℏβ[m2x˙2(τ)+V(x(τ),τ)]dτ}from where the immediate writing of the associate QS-partition function
ZQS(β)=∫−∞+∞(xbℏβ;xa0)QSdx=∫x(0)=x(ℏβ)D″xexp{−1ℏ∫0ℏβL+(x,x˙,τ)dτ}having both QS objects written as the effect of transforming the canonical Lagrangean of action into the so called Euclidian one
L+(x,x˙,τ)=m2x˙2(τ)+V(x(τ),τ)analogously with the fact the Euclidian metric has all its diagonal terms positively defined.
Yet, the connection of the path integrals of propagators with the Schrödinger quantum formalism is to be revealed next.
Schrödinger Equation from Path Integral
There are two ways for showing the propagator path integral links with Schrödinger equation.
Propagator’s Equation
Firstly, by employing one of the above path integral, say that of Equation (116) with (118)(xbtb;xatb)=∫x(ta)=xax(tb)=xbD′xexp{iℏS[x,x˙,t]}to perform the derivative
∂∂xb(xb,tb;xa,ta)=(xb+δx(tb),tb;xa,ta)−(xb,tb;xa,ta)δx(tb)=1δx(tb)[∫x(ta)=xax(tb)=xbD′xexp{iℏS[x(t)+δx(t)]}−∫x(ta)=xax(tb)=xbD′xexp{iℏS[x(t)]}]=1δx(tb)δS∫x(ta)=xax(tb)=xbD′xexp{iℏS[x(t)+δx(t)]}−∫x(ta)=xax(tb)=xbD′xexp{iℏS[x(t)]}δS=1δx(tb)δSδ(xb,tb;xa,ta)δS=1δx(tb)δSiℏ(xb,tb;xa,ta)=1δx(tb)[∫tatb(∂L∂xδx+∂L∂x˙δx˙)dt]iℏ(xa,tb;xa,ta)=1δx(tb)[∫tatb(∂L∂xδx+∂L∂x˙ddtδx)dt]1ℏ(xb,tb;xa,ta)=1δx(tb)[∂L∂x˙∂∂tδx|tatb+∫tatb(∂L∂x−ddt∂L∂x˙)︸=0Euler−Lagrangeequationδxdt]iℏ(xb,tb;xa,ta)=1δx(tb)[∂L∂x˙(tb)∂∂tδx(tb)︸≠0−∂L∂x˙(ta)∂∂tδx(ta)︸=0]iℏ(xb,tb;xa,tb)=∂L∂x˙(tb)︸p(tb)iℏ(xb,tb;xa,ta)=[iℏp(tb)](xb,tb;xa,ta)
Similarly for the second derivative we have
∂∂xb2(xb,tb;xa,ta)=[iℏp(tb)]2(xb,tb;xa,ta)=−p2(tb)ℏ2(xb,tb;xa,ta)while for time derivative we obtain
∂∂tb(xb,tb;xa,tb)=δSδtbiℏ(xb,tb;xa,ta)=−iℏH(tb)(xb,tb;xa,ta)by recalling the Hamilton-Jacobi equation of motion in the form
δSδtb=−H(tb)
Now, there is immediate that for a Hamiltonian of the form (112) one gets through multiplying both its side with the propagator (122) and then considering the relations (124) and (126), respectively, one leaves with the Schrödinger type equation for the path integral
iℏ∂∂tb(xb,tb;xa,ta)=[−ℏ22m∂∂xb2+V(xb)](xb,tb;xa,ta)
Remarkably, besides establishing the link with the Schrödinger picture, equation (127) tells something more important, namely that the wave function itself, i.e., Ψ(xb, tb), may be replaced (and generalized as well) by the quantum propagator (xb, tb; xa, ta); this has a crucial consequence since the propagator is providing the N-electronic density in the directly and elegantly manner prescribed by the algorithm (87)–(89), here actualized as
ρN(x;tb−ta)=NZ(tb;ta)(xb,tb;xa,ta)|xb=xb=xwith partition function given as in (119), assuring for the correct N-representability (DFT) constraint:
∫ρN(x;tb−ta)dx=Nthus nicely replacing the complicated many-body wave function calculations.
Nevertheless, the path integral formalism is able to provide also the exact Schrödinger equation for the wave function, as will be shown in the sequel.
Wave Function’s Equation
The starting point is the manifested equivalence between the path integral propagator and the Green function, with the role in transforming the wave-function registered on a space-time event into other one, either in the future of past quantum evolution. Here we consider only retarded phenomena modeled by the propagator
(x2,t2;x1,t1)=iG+(x2,t2;x1,t1)which in accordance with the very beginning path integral construction, the slicing (90) and the relation (91), implies the existence of the so called quantum Huygens principle of wave-packet propagation [64]
ψ(x2,t2)=∫(x2,t2;x1,t1)ψ(x1,t1)dx1,t2>t1
Yet, we will employ Equation (131) for an elementary propagator, modeling the quantum evolution presented in Figure 1, thus behaving like
ψ(x,t+ε)=A∫exp[iℏεL(x+x02,x−x02,t+ε2)]ψ(x−ξ,t)dx0where A plays the role of the normalization constant in (132a) to assure the convergence of the wave function wave-packet. Equation (132a) may be still transformed employing the geometrical relation
x=x0+ξto compute the space and velocity averages
x+x02=2x−ξ2=x−ξ2x−x0ε=ξεrespectively, while changing the variable
dx0=−dξto furnish the actual form
ψ(x,t+ε)=A˜∫exp{iℏε[m2ξ2ε2−V(x−ξ2,t+ε2)]}ψ(x−ξ,t)dξ=A˜∫exp[im2ℏεξ2]exp[−iℏεV(x−ξ2,t+ε2)]ψ(x−ξ,t)dξwhere Lagrangean was considered with its canonical form, as in (118), and the new constant factor was considered assimilating the minus sign of (133d).
Next, since noting the square dependence of ξ in (132b) there will be assumed the series expansion in coordinate (ξ) and time (ε) elementary steps restrained to the second and first order, respectively, being the time interval cut-off in accordance with the general (101) prescription. Thus we firstly have
ψ(x−ξ,t)≅ψ(x,t)−ξ[∂∂xψ(x,t)]ξ→0+ξ22[∂2∂x2ψ(x,t)]ξ→0ψ(x,t+ε)≅ψ(x,t)+ε[∂∂tψ(x,t)]ε→0exp[−iℏεV(x−ξ2,t+ε2)]≅1−iℏεV(x,t)and the form (132b) successively rearranges:
ψ(x,t)+ε[∂∂tψ(x,t)]=A˜∫e−m2iℏεξ2[1−iℏεV(x,t)]{ψ(x,t)−ξ[∂∂xψ(x,t)]+ξ22[∂2∂x2ψ(x,t)]}dξ=A˜ψ(x,t)∫e−m2iℏεξ2dξ−A˜[∂∂xψ(x,t)]∫ξe−m2iℏεξ2dξ+A˜12[∂2∂x2ψ(x,t)]∫ξ2e−m2iℏεξ2dξ−A˜iℏεV(x,t)ψ(x,t)∫e−m2iℏεξ2dξ+A˜iℏεV(x,t)[∂∂xψ(x,t)]∫ξe−m2iℏεξ2dξwhere the mixed orders producing a total order beyond the maximum equal two have been neglected, e.g., εξ2 ≅ 0, and were we arranged the exponentials under integrals of Gaussian type (i.e., employing the identity –i =1/ i). Now, the integrals appearing on (135) are of Poisson type of various orders and, assuming the notation:
m2ℏεi≡Ξare solved as:
∫e−m2iℏεξ2dξ→∫−∞+∞exp(−Ξξ2)dξ=πΞ=2πℏεim∫ξe−m2iℏεξ2dξ→∫−∞+∞exp(−Ξξ2)dξ=0∫ξ2e−m2iℏεξ2dξ→∫−∞+∞ξ2exp(−Ξξ2)dξ=12ΞπΞ=ℏεim2πℏεimWith these the expression (135) simplifies to
ψ(x,t)+ε[∂∂tψ(x,t)]=A˜2πℏεim[1+12ℏεim∂2∂x2−iℏεV(x,t)]ψ(x,t)which in the limit ε → 0, commonly for path integrals, leaves with identity
ψ(x,t)=limε→0(A˜2πℏεim)ψ(x,t)from where the convergence constant of path integral (132b) is found
A˜(ε)=m2πℏεirecovering the previous form, see Equation (114), thus confirming the consistency of the present approach. Nevertheless, with the constant (139b) back in (138) we get the equivalent forms
ψ(x,t)+ε[∂∂tψ(x,t)]=ψ(x,t)+12ℏεim∂2∂x2ψ(x,t)−iℏεV(x,t)ψ(x,t)⇔∂∂tψ(x,t)=12ℏim∂2∂x2ψ(x,t)−iℏV(x,t)ψ(x,t)⇔iℏ∂∂tψ(x,t)=[−12ℏ2m∂2∂x2+V(x,t)]ψ(x,t)being the last one identical with the Schrödinger wave function equation.
Thus it was therefore thoroughly proven that the Feynman path integral may be reduced to the quantum wave-packet motion while carrying also the information that connects coupled events across the paths’ evolution, being by all of these a general approach of quantum mechanics and statistics.
The next section will deal with presenting practical application/calculation of the path integrals for fundamental quantum systems, e.g., the free and harmonic oscillator motions.
Calculation of Path Integrals. Basic ApplicationsPath Integrals’ Properties
There are three fundamental properties most useful for path integral calculations [65].
Firstly, one may combine the two above Schrödinger type bits of information about path integrals: the fact that propagator itself (xb, tb; xa, ta) obeys the Schrödinger equation, see Equation (127), thus behaving like a sort of wave-function, and the fact that Schrödinger equation of the wave-function is recovered by the quantum Huygens principle of wave-packet propagation, see Equation (131). Thus it makes sense to rewrite Equation (131) with the propagator instead of wave-function obtaining the so called group property for propagators(x3,t3;x1,t1)=∫(x3,t3;x2,t2)(x2,t2;x1,t1)dx2,t3>t2>t1which, nevertheless, may be recursively applied until covering the entire time slicing of the interval [ta, tb] as given in (90)
(xb,tb;xa,ta)=∫(xb,tb;xn,tn)(xn,tn;xn−1,tn−1)⋯(x1,t1;xa,ta)∏j=1ndxjwhile remarking the absence of time intermediate integration.
Secondly, from the Huygens principle (131) there is abstracted also the limiting delta Dirac-function for a propagator connecting two space events simultaneously
(x,t1;x1,t1)=δ(x−x1)that is immediately proofed out
ψ(x,t1)=∫(x,t1;x1,t1)ψ(x1,t1)dx1=∫δ(x−x1)ψ(x1,t1)dx1This property is often used as the analytical check once a path integral propagator is calculated for a given system.
Thirdly, and perhaps most practically, one would like to be able to solve the path integrals, say with canonical Lagrangean form (121a), in more direct way than to consider all multiple integrals involved by the measure (117).
Hopefully, this is possible working out the quantum fluctuations along the classical path connecting two space-time events. In other words, worth to disturb the classical path xcl (t) by the quantum fluctuations δx(t) to obtain the quantum evolution path
x(t)=xcl(t)+δx(t)with its temporal derivation
x˙(t)=x˙cl(t)+δx˙(t)
Very important, note that the quantum fluctuation vanishes at the end-points of the evolution path since “meeting” with the classical (observed) path, see Figure 2δx(ta)=0=δx(tb)being these constraints known as the Dirichlet boundary conditions.
Now, one aims to separate the classical by the quantum fluctuation contributions also in the path integral propagator. Fortunately, this is possible for enough large class of potentials, more precisely for quadratic Lagrangeans of general type
L(x,x˙,t)=α(t)x2+β(t)xx˙+γ(t)x˙2+λ(t)x+χ(t)x˙+σ(t).
Actually, expanding the path integral action (118) around the classical path requires the expansion of its associate Lagrangean (146); so we get accordingly
L(x,x˙,t)=L(xcl,x˙cl,t)+[∂L∂xδx+∂L∂x˙δx˙]xcl,x˙cl+12[∂2L∂x2δx+2∂2L∂x∂x˙δxδx˙+∂2L∂x˙2δx˙2]xcl,x˙clS[x,x˙,t]=∫tatbL(x,x˙,t)dt=∫tatbL(xcl,x˙cl,t)dt+∫tatb[∂L∂xδx+∂L∂x˙δx˙]xcl,x˙cldt+∫tatb[α(t)δx+β(t)δxδx˙+γ(t)δx˙2]dt=Scl[xcl,x˙cl,t]+∂L∂x˙δx|tatb︸=0δx(ta)=0=δx(tb)+∫tatb(∂L∂x−ddt∂L∂x˙)︸=0Euler−Lagrangeδxdt+∫tatb[α(t)δx+β(t)yδx˙+γ(t)δx˙2]dtScl[xcl,x˙cl,t]+∫tatb[α(t)δx+β(t)yδx˙+γ(t)δx˙2]dt
With the action (147b) one observes it practically separates into the classical and quantum fluctuation contributions; this has two major consequences:
○ The classical action goes outside of the path integration by simply becoming the multiplication factor exp[(i / ħ)Scl];
○ Since the remaining contribution since depends only on quantum fluctuation δx(t) it allows the changing of the integration measure
∫x(ta)=xax(tb)=xbD′x(t)→∫δx(ta)=0δx(tb)=0|δx(t)δ(δx(t))|︸=1eq.(145a)D′δx(t)=∫δx(ta)=0δx(tb)=0D′δx(t)In these circumstances the path integral propagator factorizes as
(xb,tb;xa,ta)=exp{iℏScl[xcl,x˙cl,t]}︸classicalcontribution∫δx(ta)=0δx(tb)=0D′δx(t)exp{iℏ∫tatb[α(t)δx+β(t)yδx˙+γδx˙+γ(t)δx˙2]dt}︸quantumfluctutaions
Few conceptual comments are now compulsory based on the path integral form (149):
○ It is clear that the quantum fluctuation term does not depend on ending space coordinates but only on their time coordinates, so that in the end will depend only on the time difference (tb –ta) since by means of energy conservation all the quantum fluctuation is a time-translation invariant, see for instance the Hamilton-Jacobi Equation (126); therefore it may be further resumed under the fluctuation factorF(tb−ta)≡∫δx(ta)=0δx(tb)=0D′δx(t)exp{iℏ∫tatb[α(t)δx+β(t)yδx˙+γ(t)δx˙2]dt}
○ Looking at the terms appearing in the whole Lagrangean (146) and to those present on the factor (150) it seems that once the last is known for a given Lagrangean, say L, then the same is characterizing also the modified one with the terms that are not present in the forms (150), namely
L˜=L+λ(t)x+χ(t)x˙+σ(t)
○ The resulting working path integral of the propagator now simply reads
(xb,tb;xa,ta)=F(tb−ta)exp{iℏScl[xcl,x˙cl,t]}and gives intuitive inside of what path integral formalism of quantum mechanics really does: corrects the classical paths by the quantum fluctuations resumed as the amplitude of the (semi) classical wave.
Next, the big challenge is to compute the above fluctuation factor (150); here there are two possible approaches. One is considering the fluctuations as a Fourier series expansion so that directly (although through enough involving procedure) solving the multiple integrals appearing in (150). This route was originally proposed by Feynman in his quantum mechanically devoted monograph [41], and recently refined by Kleinert in an extended textbook [39].
The second way is trickier, although with limitations, but it avoids performing the direct integration prescribed by (150), while being instructive since computing the quantum fluctuation again in terms of classical path action [65], however through employing the present first two propagator properties, the group property (141) and the delta-Dirac limit (143), upon the quantum wave (152).
As such, combining the stipulated propagator properties, one starts by equivalently writing
δ(xb−xa)=(xa,t;xa,t)=∫(xb,t;x,0)(x,0;xa,t)dx=∫(xb,t;x,0)(xa,t;x,0)*dxwhere the last identity follows since using the identity between the retarded (+) and advanced (–) Green functions [64]
G+(xb,tb;xa,ta)=[G−(xa,ta;xb,tb)]*combined with the propagator-Green function relationship (130), while supplemented with the advanced propagator version
(xa,ta;xb,tb)=−iG−(xa,ta;xb,tb)
Now, the propagators from (153) may be written immediately under the general form (152)
(xb,t;x,0)=F(t)exp{iℏScl(xb,t;x,0)}(xa,t;x,0)*=F*(t)exp{iℏScl(xa,t;x,0)}contributing in rewriting (153) as
δ(xb−xa)=∫dx|F(t)|2exp{iℏ[Scl(xb,t;x,0)−Scl(xa,t;x,0)]}=Δx=xb−xaxb=xa+Δx|F(t)|2∫dxexp{iℏ[Scl(xa+Δx,t;x,0)−Scl(xa,t;x,0)]}=|F(t)|2∫dxexp{iℏ∂Scl(xa,t;x,0)∂xaΔx}=|F(t)|2∫dxexp{iℏ∂Scl(xa,t;x,0)∂xa(xb−xa)}
Next, assuming the notation
s(x)≡∂Scl(xa,t;x,0)∂xain the case its derivative ds(x) / dx is independent of x - then it goes out the integral (157) with the x-variable changed to s, leaving with the identity
δ(xb−xa)=2πℏ|F(t)|2|ds/dx|∫ds2πℏexp{iℏ(xb−xa)s}︸δ(xb−xa)from where the quantum fluctuation factor immediately follows with the analytical general form
F(t)=12πℏ|dsdx|=eq.(158)12πℏ|∂2Scl(xa,t;x,0)∂x∂xa|1/2.
With expression (160) the propagator (152) is fully expressed in terms of classical action as
(xb,tb;xa,ta)=12πℏ|∂2Scl(xa,t;x,0)∂x∂xa|1/2exp{iℏScl(xb,tb;xa,ta)},t=tb−taor in the more appealing form
(xa,tb;xa,ta)=12πℏ|∂2Scl(xb,tb;xa,tb)∂xb∂xa|1/2exp{iℏScl(xb,tb;xa,ta)}usually referred to as the Van Vleck-Pauli-Morette formula, emphasizing on the importance of solving the classical problem for a given canonical Lagrangean [60,66].
However, the path integral solution (161b) has to be used with two amendments:
○ the procedure is valid only when the quantity (158), here rewritten in the spirit of (161b) as ∂Scl (xb, tb; xa, ta)/∂xa, performed respecting one end-point coordinate remains linear in the other space (end-point) coordinate xb, so that the identity (159) holds; this is true for the quadratic Lagrangeans of type (146) but not when higher orders are involved, when the previously stipulated Fourier analysis has to be undertaken (one such case will be in foregoing sections presented).
○ In the case the formula (161b) is applicable, i.e., when previous condition are fulfilled, the obtained result has to be still verified in recovering the delta-Dirac function by the limit
limtb→ta(xb,tb,xa,ta)=δ(xb−xa)in accordance with the implemented recipe, see Equation (153); usually this step is providing additional phase correction to the solution (161b).
The present algorithm is in next exemplified on two paradigmatic quantum problems: the free motion and the motion under harmonic oscillator influence. In each case the knowledge of the classical action will almost solve the entire path integral problem.
Path Integral for Free Particle
Given a free particle with the Lagrangean
L(0)(x,x˙,t)=m2x˙2it leads by means of Euler-Lagrange equation
ddt(∂L∂x˙)=∂L∂xto the classical (Newtonian) motion
x¨cl(t)=0with the obvious solution
xcl(t)=xa+xb−xatb−ta(t−ta)fulfilling the boundary conditions
xcl(ta)=xaxcl(tb)=xbbeing these endpoints the states where the system is observable, i.e., where the quantum fluctuations vanishes, see Equation (145c) and Figure 2.
Replacing solution (165b) back in Lagrangean (163) the classical action is immediately found
S(0)cl(xb,tb;xa,ta)=∫tatbL(0)(xcl,x˙cl,t)dt=m2∫tatbx˙cl2(t)dt=m2(xb−xa)2tb−ta
Next, the quantity (158) is firstly evaluated in the spirit of (161b) as
s(0)(x)=∂S(0)cl(xb,tb;xa,ta)∂xa=mtb−ta(xb−xa)and recognized as linear in the other end-point space coordinate xb. Thus, the formula (161b) may be applied, with the actual yield
(xb,tb;xa,ta)(0)=m2πℏ(tb−ta)exp{iℏm2(xb−xa)2tb−ta}
Finally, the result (168) has to be arranged so that to satisfy the limit (162) as well. For that we use the delta-Dirac representation
δ(xb−xa)=1πlimT→0{1πexp[−(xb−xa)2T]}Comparison between (168) and (169) leads with identification
1T=−im2ℏ(tb−ta)=m2iℏ(tb−tb)thus correcting the factor of (168) towards the correct limiting path integral solution
(xb,tb;xa,ta)(0)=m2πiℏ(tb−ta)exp{iℏm2(xb−xa)2tb−ta}
Remarkably, this solution is indeed identical with the Green function of the free particle, up to the complex factor of (130), thus confirming the reliability of the path integral approach. Moreover, beside of its foreground character in quantum mechanics, the present path integral of the free particle can be further used in regaining the energy quantification of free electrons in solid state (motion within the infinite high box) as well as the Bohr quantification for the continuous deformation of the path on the circle [39,65].
Yet, these cases appeal the spectral representation of the quantum propagators and will not be treated here, being more suited for a dedicated monograph [67].
Path Integral for Harmonic Oscillator
The characteristic Lagrangean of the harmonic oscillator
L(ω)(x,x˙,t)=m2x˙2−m2ω2x2provides, when considered in the Euler-Lagrange equation (164), the classical equation of motion
x¨cl(t)+ω2xcl(t)=0with the well known solution
xcl(t)=Csin(ωt+ϕ)specialized for the end-point events of motion as
xa=xcl(ta)=Csin(ωta+ϕ)xb=xcl(tb)=Csin(ωtb+ϕ)
In the same way as done for the free motion, see solution (165b), worth rewritten the actual classical solution (173b) in terms of relations (174), for instance as
xcl(t)=Csin[ω(t−ta)+(ωta+ϕ)]=Ccos(ωta+ϕ)︸x˙a/ωsin[ω(t−ta)]+Csin(ωta+ϕ)︸xacos[ω(t−ta)]=1ωx˙asin[ω(t−ta)]+xacos[ω(t−ta)]or similarly as:
xcl(t)=1ωx˙bsin[ω(t−tb)]+xbcos[ω(t−tb)]
On the other hand the classical action of the Lagrangean (172) looks like
S(ω)cl(xb,tb;xa,ta)=∫tatbL(ω)(xcl,x˙cl,t)dt=m2∫tatb[(dxcldt)(dxcldt)−ω2xcl2]dt=m2∫tatb[ddt(xclx˙cl)−xclx¨cl−ω2xcl2]dt=[m2xcl(t)x˙cl(t)]tatb−m2∫tatb(x¨cl+ω2xcl)︸=0eq.(173a)xcldt=m2[xb(tb)x˙b(tb)−xa(ta)x˙a(ta)]
Now, in order to have classical action in terms of only space-time coordinate of the ending points, one has to replace the end-point velocities in (176) by the aid of relations (175a) and (175b) in which the current time is taken as the t = tb and t = ta, respectively; thus we firstly get
x˙a=ωsin[ω(tb−ta)]{xb−xacos[ω(tb−ta)]}x˙b=ωsin[ω(tb−ta)]{−xa+xbcos[ω(tb−ta)]}then we form the working products
xbx˙b=ωsin[ω(tb−ta)]{xb2cos[ω(tb−ta)]−xaxb}xax˙a=ωsin[ω(tb−ta)]{−xa2cos[ω(tb−ta)]+xaxb}to finally replace them in expression (176) to obtain the computed classical action
S(ω)cl(xb,tb;xa,ta)=mω2sin[ω(tb−ta)]{(xa2+xb2)cos[ω(tb−ta)]−2xaxb}
Note that the correctness of Equation (179) may also be checked by imposing the limit ω → 0 in which case the previous free motion has to be recovered; indeed by employing the consecrated limit
limy→0sinyy=1one immediately gets
limω→0S(ω)cl(xb,tb;xa,ta)=m2(tb−ta)limω→0ω(tb−ta)sin[ω(tb−ta)]︸=1limω→0{(xa2+xb2)cos[ω(tb−ta)]−2xaxb}︸(xb−xa)2=m2(xb−xa)2tb−ta=S(0)cl(xb,tb;xa,ta)
Such a kind of check is most useful and has to hold also for the quantum propagator as a whole. Going to determine it one has to reconsider the classical action (179) so that the quantity (158) is directly evaluated in the spirit of (161b) as
∂S(ω)cl(xb,tb;xa,ta)∂xa=mωsin[ω(tb−ta)]{xacos[ω(tb−ta)]−xb}thus again encountering the linearity case in the other end-point coordinate xb. Being this the fortunate situation in which the previous expression (161b) for path integral computation may be applied with the harmonic oscillator result
(xb,tb;xa,ta)=mω2πℏsin[ω(tb−ta)]exp{iℏmω2sin[ω(tb−ta)]{(xa2+xb2)cos[ω(tb−ta)]−2xaxb}}
Yet, as above was the case for the classical action itself, also the pre-exponential quantum fluctuation factor of (183a) has to overlap with that appearing in the path integral of free motion of Equation (171) for the limit ω → 0
limω→0mω2πℏsin[ω(tb−ta)]=eq.(180)m2πℏ(tb−ta)
Thus we have to adjust the propagator (183) with the exponential pre-factor corrected with the complex factor “i”
(xb,tb;xa,ta)(ω)=mω2πiℏsin[ω(tb−ta)]exp{iℏmω2sin[ω(tb−ta)]{(xa2+xb2)cos[ω(tb−ta)]−2xaxb}}
This is the sought propagator of the (electronic) motion under the harmonic oscillating potential, computed by means of path integral; it provides the canonical density to be implemented in the DFT algorithm (128), (129)
ρ⊗(x,tb−ta)=(x,tb;x,ta)(ω)=mω2πiℏsin[ω(tb−ta)]exp{iℏmω(cos[ω(tb−ta)]−1)sin[ω(tb−ta)]x2}
Yet, for practical implementations, the passage from quantum mechanics (QM) to quantum statistics (QS) is to be considered based on the Wick transformation (10) here rewritten as
(tb−ta)→−iℏβ≡−i(τb−τa)providing the Euler trigonometric to hyperbolic function conversions (by analytical continuations)
sinω(tb−ta)=1ieiω(tb−ta)−e−iω(tb−ta)2→1ieω(τb−τa)−e−ω(τb−τa)2=1isinhω(τb−τa)cosω(tb−ta)=eiω(tb−ta)+e−iω(tb−ta)2→eω(τb−τa)+e−ω(τb−τa)2=coshω(τb−τa)finally displaying for density (186) the counterpart formulation
ρ⊗(x,τb−τa)=mω2πℏsinh[ω(τb−τa)]exp{−1ℏmω(cosh[ω(τb−τa)]−1)sinh[ω(τb−τa)]x2}The uni-particle (electronic) density (189) is then used for computing the harmonic oscillator partition function
Zω=∫−∞+∞ρ⊗(x,τb−τa)dx=mω2πℏsinh[ω(τb−τa)]∫−∞+∞exp{−1ℏmω(cosh[ω(τb−τa)]−1)sinh[ω(τb−τa)]x2}dx=mω2πℏsinh[ω(τb−τa)]πℏsinh[ω(τb−τa)]mω(cosh[ω(τb−τa)]−1)=12(cosh[ω(τb−τa)]−1)Now, using the “double angle” formula
cosh2y=cosh2y+sinh2y=2cosh2y−1=2sinh2y+1the partition function (190) further becomes
Zω=12sinh[ω(τb−τa)2]=eq.(187)12sinh[ωℏβ2]
Remarkably, the result (192) recovers also the energy quantification of the quantum motion under the harmonic oscillator influence, as seen by the successive transformations
Zω=1exp(ωℏβ/2)−exp(−ωℏβ/2)=exp(−ωℏβ/2)11−exp(−ωℏβ)=exp(−ωℏβ/2)∑n=0∞[exp(−ωℏβ)]n=exp(−ωℏβ/2)∑n=0∞[exp(−nωℏβ)]=∑n=0∞exp[−βℏω(n+12)]
When comparing the expression (193a) with the canonical formulation of the partition function
Z≡∑n=0∞exp[−βEn]there follows immediately the harmonic oscillator energy quantification
En(ω)=ℏω(n+12)in perfect agreement with the consecrated expression.
The results of these two sections suggest the following rules for using path integrals propagator for density computations:
○ The reliable application of the density computation upon the partition function algorithm, see Equations (128) and (129), prescribes the transformation of the obtained quantum result to the quantum statistical counterpart by means of Wick transformation (10), while supplemented by the functions (188) conversions;
○ In computation of the path integral propagator the workable Van Vleck-Pauli-Morette formula looks like
(xb,tb;xa,ta)=12πiℏ|∂2Scl(xb,tb;xa,ta)∂xb∂xa|exp{iℏScl(xb,tb;xa,ta)}with the complex factor “i” included, as confirmed by both the free and harmonic oscillator quantum motions; it may be used for linear classical actions in one of the end-point space coordinates upon derivation respecting the other one; yet the formula (195) should be always checked for fulfilling the limiting (162) delta-Dirac function for simultaneous events for any applied potential.
Nevertheless, recognizing the major role the classical action plays in the path integral representation of the quantum propagation (and propagator), the question whether it is possible to consider the semi-classical expansion of the propagator in general case, without being under any constraint except the semiclassical (higher temperatures) limit itself, naturally arises. Such an approach is exposed and its reliability tested in the next sections.
Semiclassical Path Integral of Evolution AmplitudeSemiclassical Expansion
Semiclassical derivation of the evolution amplitude employs some of the previously Feynman path integral ideas refined due to the works of Kleinert and collaborators [39,41,68–71]. They are bellow summarized.
○ The real time dependency is “rotated” into the imaginary time
τ=it=ℏβor in finite differences as
τb−τa=ℏβ=i(tb−ta)according with the Wick transformation (10).
○ The quantum paths of (145a) are re-parameterized as
x(τ)=x¯+η(τ)where the classical path of (145a) is replaced by the fixed (non time-dependent) average
xcl(t)→x¯=xa+xb2while the fluctuation path η(τ) remains to carry the whole path integral information, yet being departed at the end of integration frontier from previously Dirichlet boundary conditions (145c), where it vanished at the domain frontiers, to the actual different endpoint values
ηa=η(τa)=−Δx2ηb=η(τb)=+Δx2in terms of the length of the “traveled” space
Δx=xb−xa
In these conditions the quantum statistical path integral representation of quantum propagator becomes
(xbτb;xaτa)QS=∫η(τa)=−Δx/2η(τb)=+Δx/2Dη(τ)exp{−1ℏ∫τaτb[m2η˙2(τ)+V(x¯+η(τ))]dτ}since we immediately noted the immediate transformations
D′x(τ)→Dη(τ)x˙2(τ)→η˙2(τ)V(x(τ))→V(x¯+η(τ))based on the above (197)–(200) parameterization.
It should be pointed out that the used re-parameterization is not modifying the value of the path integral but is intended to better visualize its properties, towards evaluating it. As such, from expression (201) it now appears clearer than before that for the systems governed by smooth potentials, the series expansion may be applied respecting the path fluctuation, here in the second order truncation
V(x¯+η(τ))≅V(x¯)+∂iV(x¯)ηi(τ)+12∂i∂jV(x¯)ηi(τ)ηj(τ)where the covariant notation for products was assumed for maintaining the generality of the D-dimensioned approach. This way, a (truncated) series of path integral evolution amplitude of Equation (201) it is at once obtained
(xbτb;xaτa)QS=∫η(τa)=−Δx/2η(τb)=+Δx/2Dη(τ)e−1ℏ∫τaτb[m2η˙2(τ)]dτe−1ℏ∫τaτbV(x¯)dτe−1ℏ∫τaτb[∂iV(x¯)ηi(τ)+12∂i∂jV(x¯)ηi(τ)ηj(τ)+…]dτ=exp[−τb−τaℏV(x¯)]∫η(τa)η(τb)Dη(τ)exp{−1ℏ∫τaτb[m2η˙2(τ)]dτ}×{1−1ℏ∫τaτbdτ[∂iV(x¯)ηi(τ)+12∂i∂jV(x¯)ηi(τ)ηj(τ)+…]+12ℏ2∫τaτbdτ∫τaτbdτ′[∂iV(x¯)∂jV(x¯)ηi(τ)ηj(τ′)+…]+…}as being driven by the quantum fluctuation’ various orders contributions, up to the second order. This is a natural approach since the very quantum nature of the path integral is given by the quantum fluctuations themselves, from where the systematic approximations of path integrals over the quantum fluctuations. The series is known as the semiclassical expansion since is formally done in the “powers of ħ-Planck”.
Now, looking on Equation (204) as compared with the previously used quantum mechanical form of Equation (149) the present propagator representation would be resumed as
(xbτb;xaτa)QS=exp[−βV(x¯)]︸CLASSICALFACTOR∫η(τa)η(τb)Dη(τ)FSC[η]exp{−1ℏ∫τaτb[m2η˙2(τ)]dτ}︸SEMI−CLASSICALCONTRIBUTIONwhere we have used (196b) along identifying the semiclassical factor FSC[η]. However, the expression (205a) may be further formally cast
(xbτb;xaτa)QS=exp[−βV(x¯)](Δx2τb;−Δx2τa)(0)〈FSC[η]〉by introducing the so called free imaginary time amplitude
(Δx2τb;−Δx2τa)(0)=∫η(τa)=−Δx/2η(τb)=+Δx/2Dη(τ)exp{−1ℏ∫τaτb[m2η˙2(τ)]dτ}readily given by the free-propagator solution (171) accommodated by the present statistical and boundary transformations Equations (196b) and (199) to the form
(Δx2τb;−Δx2τa)(0)=m2πℏ2βexp{−m2ℏ2β(Δx)2}while having the normalization role for averaging the semiclassical factor contribution
〈FSC[η]〉=∫η(τa)=−Δx/2η(τb)=+Δx/2Dη(τ)FSC[η]exp{−1ℏ∫τaτb[m2η˙2(τ)]dτ}(Δx2τb;−Δx2τa)(0)
Therefore, the semiclassical form of path integral representation of evolution amplitude looks like
(xbτb;xaτa)QS=m2πℏ2βexp{−m2ℏ2β(Δx)2−βV(x¯)}×{1−1ℏ∂iV(x¯)∫τaτbdτ〈ηi(τ)〉−12ℏ∂i∂jV(x¯)∫τaτbdτ〈ηi(τ)ηj(τ)〉+12ℏ2∂iV(x¯)∂jV(x¯)∫τaτbdτ∫τaτbdτ′〈ηi(τ)ηj(τ′)〉+...}
The remaining problem is that of expressing the averaged values of the fluctuation paths in single or multiple time connection, i.e., 〈ηi(τ)〉, 〈ηi(τ)ηj(τ)〉, 〈ηi(τ)ηj(τ')〉, etc.
From the heuristic point of view it is normal to arrive at the form (208) because it tells us that the quantum fluctuation is firstly averaged along the quantum evolution and then averaged by time in order that the evolution amplitude is determined. Observe also that the present semiclassical approach is not using the previously employed properties of the classical action, avoiding therefore the limitation of the derivative behavior at edge of the space domain of integration, while posing now the limitation in what respect the quantum fluctuation power. It is also useful to remark that the present semiclassical approach may use the interplay between the previous solved free-and-harmonic quantum motions, since the path integral (206a) may equally be regarded as the free motion of the quantum fluctuations (naturally since they are not known a priori or with some possibility of instantaneously observation); at the same time, if one formally counts the kinetic term as the perturbative (a.k.a. fluctuation) oscillatory motion
m2η˙2(τ)=m2(ddtη)2≈m2ω2︸∂tη2a more complex picture of quantum fluctuation is obtained; in conclusion, quantum fluctuating paths may be (or should be) treated as being a kind of harmonically free motion: harmonic since as fluctuations may be expanded in Fourier series (as originally perceived by Feynman), but also free since their unknown of instantaneous feature. Therefore, an appropriate use of both these manifestations will conduct to the reliable path integral representation. Yet, since we have already used the free-motion character of fluctuation paths, the harmonic one is next entering the analysis.
Connected Correlation Functions
For calculating the average of quantum fluctuation paths one has to understand their inner nature: in order reconciliation of free and harmonic features be achieved the so called quantum current j(τ) is introduced (and presumed to appear in reality too as causing/driving the quantum fluctuations), so that the propagator of this current, known as the generating functional, is formed [39,66]
(Δx2τb;−Δx2τa)[j(τ)]=∫η(τa)=−Δx/2η(τb)=+Δx/2Dη(τ)exp{−1ℏ∫τaτb[m2η˙2(τ)−j(τ)η(x)]dτ}with which help one can recognized the equivalence
〈η(τ)〉=ℏδδj(τ)(Δx2τb;−Δx2τa)[j(τ)]|j(τ)=0(Δx2τb;−Δx2τa)(0)=∫η(τa)=−Δx/2η(τb)=+Δx/2Dη(τ)[η(x)]exp{−1ℏ∫τaτb[m2η˙2(τ)−j(τ)η(x)]dτ}(Δx2τb;−Δx2τa)(0)in accordance with general definition (207). One can nevertheless see that the quantum current appearance in (211) is under the perturbation form, so that it readily accounts for the deviation from the free fluctuation motion towards the harmonically one. Therefore, although general correlation definition may be advanced by the ordering rule
〈η(τ1)⋯η(τn)〉=ℏδδj(τ1)⋯ℏδδj(τn)(Δx2τb;−Δx2τa)[j(τ1),⋯,j(τn)]|j(τ1)=0,...j(τn)=0(Δx2τb;−Δx2τa)(0)the problem of practically evaluation still remains. Aiming for solving it one observes that the form (212) is analogous with the partition function based electronic density, see Equation (12) for instance; consequently, the alternative formulation looks at the canonical (N = 1, mono-particle) level like
〈η(τ1)⋯η(τn)〉={Z−1[j]ℏδδj(τ1)⋯ℏδδj(τn)Z[j]}j=0where, now, the quantity Z[j] plays the role of the generating functional of the quantum fluctuation correlation (or connection) average. Yet, the writing (213) may suffer from disconnecting character due to the presence of simple Z[j]; this may be better visualized when re-expressing (213) under the so called n-point (correlation) functions〈η(τ1)⋯η(τn)〉=Z−1∫Dη(τ)[η(τ1)⋯η(τn)]exp{−1ℏS+}=∏i=1n[∫−∞+∞dη(τi)](ηbτb;ηnτn).η(τn).(ηnτn;ηn−1τn−1).η(τn−1)⋯η(τ1)(η1τ1;ηaτa)with S+ being the Euclidian action, see Equation (121c), while space and time slicing intervals are those introduced by (81) and (90), respectively.
The disconnected character of correlations (213)–(214) may be overcome remembering that the logarithm of the partition function provides the thermodynamic free energy, see relation (67), here under canonical (N = 1) form
lnZ[j]=−βF[j]which, as a measurable-observable energy, it compulsory contains the connected parts of Z[j](i.e., energy’s pieces combines towards the total energy). Therefore, this leaves with the idea that through introducing another generating functional, namely
W[j]=lnZ[j]
Equation (213) may be rewritten as the connected part of correlation 〈η(τ1)⋯η(τn)〉 that it can be naturally identified with a sort of generalized n-points (events) Green function
Gcon(n)(τ1,⋯,τn)={δδj(τ1)⋯δδj(τn)W[j]}j=0
Nevertheless, aiming to have a better “feeling” on how the connected and disconnected correlation (fluctuation) functions (217) and (213) are linked, let’s start evaluating some orders of them.
As such, absorbing the constants in the involved functionals, the first order of (213) reads correlation of (213) we successively have
〈η(τ)〉=Z−1[j]δδj(τ)Z[j]=δδj(τ)W[j]=Gcon(1)(τ)=ηcl(τ)as the single connected path (remember that the fluctuation was already averaged out) that is just the classical path connecting the ending points of the quantum evolution, see Figure 2.
Now, going to the second order of correlation of (213) one has
〈η(τ1)η(τ2)〉=Z−1[j]δδj(τ1)δδj(τ2)Z[j]=Z−1[j]δδj(τ1){Z[j](δδj(τ2)W[j])}=δδj(τ1)δδj(τ2)W[j]+Z−1[j]δZ[j]δj(τ1)δW[j]δj(τ2)=δδj(τ1)δδj(τ2)W[j]+δW[j]δj(τ1)δW[j]δj(τ2)a result that can be wisely rearranged as
〈η(τ1)η(τ2)〉=Gcon(2)(τ1,τ2)+Gcon(1)(τ1)Gcon(1)(τ2)=Gcon(2)(τ1,τ2)︸connectedevents+〈η(τ1)〉〈η(τ2)〉︸dis–connectedeventsor, even more practically for our purpose, as
〈η(τ1)η(τ2)〉=Gcon(2)(τ1,τ2)+ηcl(τ1)ηcl(τ2)
In similar manner, while applying a kind of recursive rule, sometimes denoted as the cluster decomposition or cumulant expansion〈η(τ1)⋯η(τn)〉=Gcon(n)(τ1,⋯,τn)+〈η(τ2)⋯η(τn)〉Gcon(1)(τ1),n≥2while involving the pair-wise (Wick) decomposition of the n-points correlated function
Gcon(n)(τ1,⋯,τn)=∑j=2nGcon(2)(τ1,τj)〈η(τk)⋯η(τ1)〉︸(n−2)orderfunctionk≠1,jl≠1,j,kone can easily obtain the higher orders of correlations, however observing that all connected orders of events reduce to the combinations of pair-connected events. For instance, we get for the third order fluctuations the average contribution
〈η(τ1)η(τ2)η(τ3)〉=〈η(τ1)〉〈η(τ2)η(τ3)〉+Gcon(2)(τ1,τ2)〈η(τ3)〉+Gcon(2)(τ1,τ3)〈η(τ2)〉=ηcl(τ1)[ηcl(τ2)ηcl(τ3)+Gcon(2)(τ2,τ3)]+Gcon(2)(τ1,τ2)ηcl(τ3)+Gcon(2)(τ1,τ3)ηcl(τ2)=ηcl(τ1)ηcl(τ2)ηcl(τ3)+ηcl(τ1)Gcon(2)(τ2,τ3)+ηcl(τ2)Gcon(2)(τ1,τ3)+ηcl(τ3)Gcon(2)(τ1,τ2)or with more terms involved for the fourth order
〈η(τ1)η(τ2)η(τ3)η(τ4)〉=〈η(τ1)〉〈η(τ2)η(τ3)η(τ3)〉+Gcon(2)(τ1,τ2)〈η(τ3)η(τ4)〉+Gcon(2)(τ1,τ3)〈η(τ2)η(τ4)〉+Gcon(2)(τ1,τ4)〈η(τ2)η(τ3)〉=ηcl(τ1)[ηcl(τ1)ηcl(τ2)ηcl(τ3)+ηcl(τ1)Gcon(2)(τ2,τ3)+ηcl(τ2)Gcon(2)(τ1,τ3)+ηcl(τ3)Gcon(2)(τ1,τ2)]+Gcon(2)(τ1,τ2)[Gcon(2)(τ3,τ4)+ηcl(τ3)ηcl(τ4)]+Gcon(2)(τ1,τ3)[Gcon(2)(τ2,τ4)+ηcl(τ2)ηcl(τ4)]+Gcon(2)(τ1,τ4)[Gcon(2)(τ2,τ3)+ηcl(τ2)ηcl(τ3)]=ηcl(τ1)ηcl(τ2)ηcl(τ3)ηcl(τ4)+ηcl(τ1)ηcl(τ2)Gcon(2)(τ3,τ4)+ηcl(τ1)ηcl(τ3)Gcon(2)(τ2,τ4)+ηcl(τ1)ηcl(τ4)Gcon(2)(τ2,τ3)+ηcl(τ2)ηcl(τ3)Gcon(2)(τ1,τ4)+ηcl(τ2)ηcl(τ4)Gcon(2)(τ1,τ3)+ηcl(τ3)ηcl(τ4)Gcon(2)(τ1,τ2)+Gcon(2)(τ1,τ2)Gcon(2)(τ3,τ4)+Gcon(2)(τ1,τ3)Gcon(2)(τ2,τ4)+Gcon(2)(τ1,τ4)Gcon(2)(τ2,τ3)
Next, having these examples in hand, one tries to re-deriving them by an appropriate generating functional (210) worked with the connected function definition (212). At this moment one uses the previously emphasized “free-harmonic motion” dual nature of fluctuation paths – see Equation (209), to reconsider the free imaginary time amplitude (206a) contribution
(Δx2τb;−Δx2τa)(ω)=∫η(τa)=−Δx/2η(τb)=+Δx/2Dη(τ)exp{−1ℏSω+(η,η˙,τ)}with harmonic Euclidian action of fluctuations
Sω+(η,η˙,τ)=∫τaτb[m2η˙2(τ)+m2ω2η2(τ)]dτwhile fulfilling (at the end of calculation) the constraint
limω→0(Δx2τb;−Δx2τa)(ω)=(Δx2τb;−Δx2τa)[j]
We like to rearrange the action (222b) so that the quantum current contribution to clearly appear; in achieving this one firstly rewrites it by performing the integration by parts
Sω+(η,η˙,τ)=∫τaτb{m2[ddτ(ηdηdτ)−ηd2ηdτ2]+m2ω2η2(τ)}dτ=m2(ηη˙)τaτb︸0+∫τaτb{m2η(τ)[−∂τ2+ω2]η(τ)}dτ=∫τaτb{m2η(τ)Dω(τ)η(τ)}dτwhere we have recognized the harmonic differential operator
Dω(τ)=−∂τ2+ω2The form (223) is very useful through employing the Green equation of harmonic motion
Dω(τ)Gω(τ,τ′)=δ(τ−τ′)for the integral property
∫Dω(τ)Gω(τ,τ′)f(τ,τ′)dτ=∫f(τ,τ′)δ(τ−τ′)dτ=f(τ′,τ′)to perform the path shifting of fluctuations by the transformation
η(τ)→η′(τ)=η(τ)+1mGω(τ,τ′)j(τ)on the combined term
m2η(τ)Dω(τ)η(τ)→m2η′(τ)Dω(τ)η′(τ)−[∫η′(τ)Dω(τ)Gω(τ,τ′)j(τ′)]dτ′+12m∬[j(τ)Dω(τ)Gω(τ,τ′)Gω(τ′,τ″)j(τ″)]dτ′dτ″so that the prescribed action of (223) takes the form
Sω+(η,η˙,τ)→∫τaτb{m2η′(τ)Dω(τ)η′(τ)}dτ−∫τaτbη′(τ′)j(τ′)dτ′+12m∫τaτb∫τaτbj(τ′)Gω(τ′,τ″)j(τ″)dτ′idτ″iunder the assumption the physical integration interval (τa,τb) assimilates the entirely evolution universe of the concerned problem, i.e., the mathematical interval (–∞,+∞), so that the delta-Dirac integration property (225b) is consistent. Also note that in expression (228) since the statistical Green function should come from its associate original real time quantum mechanically problem, see below, it tracks also the temporal Wick “rotation t = τ/i ” in the integration measure, explaining therefore the complex factors in the last term of (228).
With these, the harmonic fluctuation action of (228) may be reconsidered with the working form
S˜ω+(η,η˙,τ)=∫τaτb{m2η(τ)[−∂τ2+ω2]η(τ)}dτ−∫τaτbη(τ)j(τ)dτ−12m∫τaτb∫τaτbGω(τ,τ′)j(τ)j(τ′)dτdτ′such that it can be further rearranged so that the free terms action to appear distinctively under the condition ω → 0 as
S˜ω→0+(η,η˙,τ)=S0+(η,η˙,τ)−∫τaτbη(τ)j(τ)dτ−12m∫τaτb∫τaτbGω→0(τ,τ′)j(τ)j(τ′)dτdτ′with the free Euclidian action
S0+(η,η˙,τ)=∫τaτb[m2η˙2(τ)]dτrecovered though considering on expression (229) the reverse integration by parts - as unfolded from (222b) to (223).
Finally, back with the identifications
η(τ)→ηcl(τ)ℏmGω→0(τ,τ′)→Gcon(2)(τ,τ′)in the action (230), there follows that its last two terms are no longer displaying quantum fluctuations upon integration since they were comprised under averaged forms, see Equations (218) and (219c), respectively; thus they release for the searched current-dependent amplitude of (222c) the actual solution
(Δx2τb;−Δx2τa)[j]=limω→0∫η(τa)=−Δx/2η(τb)=+Δx/2Dη(τ)exp{−1ℏSω+(η,η˙,τ)}=∫η(τa)=−Δx/2η(τb)=+Δx/2Dη(τ)exp{−1ℏS˜ω→0+(η,η˙,τ)}=(∫η(τa)η(τb)Dη(τ)exp{−1ℏS0+(η,η˙,τ)})×exp(1ℏ∫τaτbηcl(τ)j(τ)dτ+12ℏ2∫τaτb∫τaτbGcon(2)(τ,τ′)j(τ)j(τ′)dτdτ′)which ultimately simply re-writes like the current dependent propagator amplitude
(Δx2τb;−Δx2τa)[j]=(Δx2τb;−Δx2τa)(0)exp(1ℏ∫τaτbηcl(τ)j(τ)dτ+12ℏ2∫τaτb∫τaτbGcon(2)(τ,τ′)j(τ)j(τ′)dτdτ′)as a wave-perturbative form of the free fluctuation amplitude (206a), being intermediated by the harmonic towards free limiting motion of quantum fluctuations. Let’s further comment that the actual form (232b) generalizes the previously “guessed” form (210), which provided the first order fluctuation correlation, however having in addition the power to recover all other superior orders of correlation, for instance those given by Equations (219c) and (221), by successive application of the formula (212); this way, the transformation factor ħ/ m in (231b) is as well justified.
With these the connected correlation function algorithm was proofed in detail, being at disposition to be implemented for whatever order of semiclassical expansion of the path integral evolution amplitude (208); as exemplification, the next section will expose the analytic solution for the second order case.
Classical Fluctuation Path and Connected Green Function
We have already seen that aiming to evaluate any of the above connected correlation functions one imperatively needs to know the analytical forms of classical fluctuation path
〈ηi(τ)〉=η(cl)i(τ)and for the connected Green function - identified from (231b) as
Gcon(2)(τ,τ′)=ℏmGω→0(τ,τ′)while depending on its turn of the knowledge of the Green function for the harmonic oscillator problem, Equation (225a) with (224), to be finally specialized towards the “free harmonic” limit ω → 0.
Therefore, with the quantities of Equations (233) any semiclassical problem can be solved analytically. Yet, for the quantum objects in question the computing procedure consists by three major stages, as unfolded in the sequel.
Solving the associate real time harmonic problem;
Rotating the solution into imaginary time picture;
Taking the “free harmonic limit”ω → 0.
Calculation of Classical Fluctuation Path
As discussed above, the classical path for quantum fluctuation will not be written directly from the ordinary path free motion (Section 3.3.2) but using the similar result of harmonic motion (Section 3.3.3) upon which the free-harmonic condition ω → 0 will be imposed; actually, the procedure is unfolded as follows.
The result (175a) is combined with (177a) to provide the real time classical path
x(ω)cl(t)=1ωωsin[ω(tb−ta)]{xb−xacos[ω(tb−ta)]}sin[ω(t−ta)]+xacos[ω(t−ta)]=xbsin[ω(t−ta)]+xa{sin[ω(tb−ta)]cos[ω(t−ta)]−cos[ω(tb−ta)]sin[ω(t−ta)]}sin[ω(tb−ta)]=xbsin[ω(t−ta)]+xasin[ω(tb−t)]sin[ω(tb−ta)]
The real to imaginary time rotation is performed on the result (234) according with the Wick rule prescription of (196a), being this equivalently of directly rewriting of expression (234) replacing the trigonometric functions by their hyperbolic counterparts, according with the previously explained conversion, see Equation (188a)xcl(τ)=xbsinh[ω(τ−τa)]+xasinh[ω(τb−τ)]sinh[ω(τb−τa)]
The “free-harmonic” (ω → 0) limit is performed upon the expression (235) through employing the ordinary hyperbolic limit
limω→0sinh[ωΞ]=ωΞ
This gives
x(ω→0)cl(τ)=xb(τ−τa)+xa(τb−τ)τb−τawhich evidently does the same job as the classical free-motion result of (165b), although not identical, since derived from a generalized perspective here.
The result (237) is implemented in the formula (197) to finally produce the classical fluctuation path
〈η(τ)〉=ηcl(τ)≡η(ω→0)cl(τ)=x(ω→0)cl(τ)−x¯=xb(τ−τa)+xa(τb−τ)τb−τa−xa+xb2=[ττb−τa−12τb+τaτb−τa]Δxwhich takes even the simpler form
〈η(τ)〉=(τℏβ−12)Δxwhen rewritten within the thermodynamic picture
τa=0τb=ℏβ
Calculation of the Connected Green Function
Now, going to the evaluation of the expression (233b) we need the Green function of the harmonic oscillator from the equation (225a); when rewritten in real time picture
[−∂t2+ω2]Gω(t,t′)=δ(t−t′),t>t′∈(ta,tb)it has the advantage of having the frontier values fixed by the Dirichlet boundary conditions
Gω(t,t′)=0{t=tb,∀t′∀t,t′=tain the same manner as the fluctuation paths in real time are set to vanish at the endpoint frontier, see Equation (145c). Such double boundary condition fixes the type of solution as being of the double trigonometric form
Gω(t,t′)=Csin[ω(tb−t)]sin[ω(t′−ta)]
In the same manner the temporal alternative ordering problem of (240)
[−∂t′2+ω2]Gω(t′,t)=δ(t′−t),t′>t∈(ta,tb)with the Dirichlet boundary conditions
Gω(t′,t)=0{t′=tb,∀t∀t′,t=taproduces the variant Green function
Gω(t′,t)=Csin[ω(tb−t′)]sin[ω(t−ta)]with the same constant as for the solution (240c) since recognizing that both formally belong to the same homogeneous equation of type
[−∂t>2+ω2]Gω(t>,t<)=0,t>>t<∈(ta,tb)
Now, looking for appropriate identification in the inhomogeneous equation
−∂t2Gω(t,t′)=δ(t−t′)+ω2Gω(t,t′)one notes that its left side is formed from the difference of the first derivatives of the solutions (240c) and (241c) approaching each other for the concerned times
−∂t2Gω(t,t′)=−limt→t′∂tGω(t,t′)−∂tGω(t′,t)t−t′=Cωlimt→t′1t−t′{cos[ω(tb−t)]sin[ω(t′−ta)]+sin[ω(tb−t′)]cos[ω(t−ta)]}=Cω{cos[ω(tb−t′)]sin[ω(t′−ta)]+sin[ω(tb−t′)]cos[ω(t′−ta)]}limt→t′1t−t′=Cωsin[ω(tb−ta)]δ(t−t′)which, through comparing with the right side first term of (243) gives the searched constant
C=1ωsin[ω(tb−ta)]
It leaves with the real time Green function solution of the harmonic oscillator
Gω(t,t′)=Θ(t−t′)sin[ω(tb−t)]sin[ω(t′−ta)]+Θ(t′−t)sin[ω(tb−t′)]sin[ω(t−ta)]ωsin[ω(tb−ta)]that nevertheless combines both above solution with the help of Heaviside step-function
Θ(t−t′)={1,t>t′12,t=t′0,t<t′
Next, as previously done with the fluctuation paths, the change to the imaginary time picture is done automatically through trigonometric-to-hyperbolic recipe (188a) to give
Gω(τ,τ′)=Θ(τ−τ′)sinh[ω(τb−τ)]sinh[ω(τ′−τa)]+Θ(τ′−τ)sinh[ω(τb−τ′)]sinh[ω(τ−τa)]ωsinh[ω(τb−τa)]noting that in the course of transformation the factor
i(i1i⋅i)=1was tacitly absorbed, with the parenthesis complex indices coming from the trigonometric to hyperbolic rotation (188a), while the outside index complex assures the equivalence of Green function contribution for the canonic-to-Euclidian path integrals action exponents’ transformation
exp(iℏ[⋯Gω(t,t′)⋯])︸CANONIC→exp(iℏ[⋯iGω(t→τ,t′→τ′)⋯])=exp(−1ℏ[⋯Gω(τ,τ′)⋯])︸EUCLIDIAN
Expression (248) is finally employed to the “free harmonic” limit (236) providing the result
Gω(τ,τ′)=Θ(τ−τ′)(τb−τ)(τ′−τa)+Θ(τ′−τ)(τb−τ′)(τ−τa)τb−τawhich being free of harmonic influence it remains identically also from the quantity Gω→0 (τ, τ'). Still, it has to be converted into the searched connected Green function (233b), leaving with the time imaginary form
Gcon(2)(τ,τ′)=ℏmΘ(τ−τ′)(τb−τ)(τ′−τa)+Θ(τ′−τ)(τb−τ′)(τ−τa)τb−τaor with its equivalent statistical one
Gcon(2)(τ,τ′)=Θ(τ−τ′)(ℏβ−τ)τ′+Θ(τ′−τ)(ℏβ−τ′)τmβwhen the thermodynamical picture (239) is considered.
Second Order Semiclassical Propagator, Partition Function and Density
Returning to evaluate the second order truncated expansion (208) one needs the evaluation of the quantities 〈ηi(τ)〉, 〈ηi(τ)ηj(τ)〉, 〈ηi(τ)ηj(τ')〉 and of their integration. Given the previous discussions, see for instance the Equations (218) and (219c), one immediately has
〈ηi(τ)〉=η(cl)i(τ)〈ni(τ)ηj(τ′)〉=η(cl)i(τ)η(cl)j(τ′)+Gcon(2)(τ,τ′)δij
With the help of expression (238) the first order averaged fluctuation integral appearing on (208) becomes
∫τaτb〈η(τ)〉dτ=∫τaτbηcl(τ)dτ=0
Going now to the double connected correlation functions, one has the working analytical expression
〈ηi(τ)ηj(τ′)〉=(ττb−τa−12τb+τaτb−τa)(τ′τb−τa−12τb+τaτb−τa)ΔxiΔxj+ℏmΘ(τ−τ′)(τb−τ)(τ′−τa)+Θ(τ′−τ)(τb−τ′)(τ−τa)τb−τaδijobtained by replacing into the expression (252b) the classical fluctuation paths and the connected Green function components, Equations (238a) and (251a), respectively.
Now, the second order averaged fluctuation integrals are computed as following:
At coincident times
∫τaτb〈ηi(τ)ηj(τ)〉dτ=δijℏm(τb−τa)∫τaτb(τb−τ)(τ−τa)dτ+ΔxiΔxj∫τaτb(ττa−τb−12τb+τaτb−τa)2dτ=δijℏm(τb−τa)26+ΔxiΔxjτb−τa12or re-written as
∫0ℏβ〈ηi(τ)ηj(τ)〉dτ=δijℏ3β26m+ΔxiΔxjℏβ12within the thermodynamical environment given by Equation (239).
At different times
∫τaτbdτ∫τaτbdτ′〈ηi(τ)ηj(τ′)〉=δijℏm(τb−τa)[∫τaτbdτ∫τaτdτ′(τb−τ)(τ′−τa)+∫τaτbdτ′∫τaτ′dτ(τb−τ′)(τ−τa)]+[∫τaτb(ττb−τa−12τb+τaτb−τa)dτ][∫τaτb(τ′τb−τa−12τb+τaτb−τa)dτ′]ΔxiΔxj=δijℏm(τb−τa)312with its quantum thermodynamical counterpart
∫τaτbdτ∫τaτbdτ′〈ηi(τ)ηj(τ′)〉=δijℏ4β312m
Finally, while replacing the values of Equations (253), (255b), and (256b) back in the second order truncated semiclassical expression of imaginary time amplitude (208) one gets the analytical result
(xbℏβ;xa0)QS[II]=m2πℏ2βexp{−m2ℏ2β(Δx)2−βV(x¯)}×{1−1ℏ∂iV(x¯)∫0ℏβdτ〈ηi(τ)〉−12ℏ∂i∂jV(x¯)∫0ℏβdτ〈ηi(τ)ηj(τ)〉+12ℏ2∂iV(x¯)∂jV(x¯)∫0ℏβdτ∫0ℏβdτ′〈ηi(τ)ηj(τ′)〉}=m2πℏ2βexp{−m2ℏ2β(Δx)2−βV(x¯)}×{1−12ℏ∂i∂jV(x¯)[δijℏ3β26m+ΔxiΔxjℏβ12]+12ℏ2∂iV(x¯)∂jV(x¯)δijℏ4β312m};(xbℏβ;xa0)QS[II]=m2πℏ2βexp{−m2ℏ2β(Δx)2−βV(x¯)}×{1−ℏ2β212m∇2V(x¯)−β24(Δx∇)2V(x¯)+ℏ2β324m[∇V(x¯)]2}
Note that the expression (257) plays the role of the semiclassical canonical density in PI-DFT algorithm given by Equations (128) and (129)ρ⊗QS[II](x¯,β)=(xbℏβ;xa0)QS[II]|xa=xb=m2πℏ2β{1−ℏ2β212m∇2V(x¯)+ℏ2β324m[∇V(x¯)]2}exp{−βV(x¯)}to construct the N-body density at thermodynamic equilibrium
ρN−QS[II](x¯,β)=NZ[II](β)ρ⊗QS[II](x¯,β)by means of partition function
Z[II](β)=∫ρ⊗QS[II](x¯,β)dx¯=m2πℏ2β∫{1−ℏ2β212m∇2V(x¯)+ℏ2β324m[∇V(x¯)]2}exp{−βV(x¯)}dx¯
At this point, the expression (260) may be elegantly transformed through considering the Gauss theorem of integrated divergence that written in a general D-dimensional case
∫∇{[∇V(x¯)]exp[−βV(x¯)]}dDx¯=0leaves with the useful differential relationship
∫[∇V(x¯)]2exp[−βV(x¯)]dDx¯=1β∫∇2V(x¯)exp[−βV(x¯)]dDx¯helping in rearranging the partition function (260) firstly as
Z[II](β)=m2πℏ2β∫[1−ℏ2β224m∇2V(x¯)]exp{−βV(x¯)}dx¯and finally, after the exponential resume, equivalently as
Z[II](β)=m2πℏ2β∫exp[−βV(x¯)−ℏ2β224m∇2V(x¯)]dx¯
In the same manner also the higher orders of semiclassical expansion of density matrix (204) or (208) can be constructed by following the cumulant expansion (220), its fluctuation path and connected Green function components, as given by Equations (238) and (251), respectively, towards producing the analytical canonical density, the partition function and finally the many-body density to be used in density functional theory and of its (chemical) applications [71]. Such an application is to be in next presented for electronegativity and chemical hardness indices’ computations.
Fourth Order Semiclassical Electronegativity and Chemical Hardness
Here, we assess electronegativity (EN) as the convolution of the imaginary time conditional probability (r, τ|0,0), τ = Im(it), with the radial valence shell potentialV (r) [72]
χ=∫V(r)(r,τ;0,0)drso representing the power of the entire atom (nucleus + core electrons + valence shell) to attract electrons of the outer shell (fixed by radius r) to its center (r = 0). This way, the current EN definition is refined to account for the whole stability of the atom with its electronic and nuclear subsystems.
Having an analytical EN quantum formulation, the chemical hardness, η, its natural companion, is re-expressed from Equation (13b) under the explicitly working form [73]
η≡−(∂χ∂N)V(r)playing a major role in establishing the main chemical principles of reactivity towards stability: the hard-and-soft-acids-and-bases (HSAB) and the maximum hardness (MH) [74–78].
The general radial one-dimensional probability amplitude connecting the space-time events (ra, 0) and (rb, τ) for electronic evolution amplitude in an atom can be derived from the semiclassical expansion, by extending the expansion (208) up to the fourth order through including the terms of type (221), whose calculation leads to the result [71]
(rb,τ;ra,0)=m2πℏτexp{−m2ℏτ(Δr)2−τℏV(r¯)}×{1−1ℏ[12V″(r¯)(τ12Δr+ℏmτ26)+124V″″(r¯)(τ80(Δr)4+ℏmτ220(Δr)2+ℏ2m2τ310)]+12ℏ2[ℏmV′(r¯)2τ312+14V″(r¯)2(τ2144(Δr)4+ℏmτ330(Δr)2+ℏ2m2τ420)+V‴(r¯)V′(r¯)(ℏmτ3240(Δr)2+ℏ2m2τ460)]−16ℏ3[32V″(r¯)V′(r¯)2(ℏmτ4144(Δr)2+11360ℏ2m2τ5)]+11152ℏ4V′(r¯)4ℏ2m2τ6}Further on, we recognize that such density matrix becomes uniformly in the valence shell radius variable rb by fixing the atomic origin in rara=0⇒{r¯=ra+rb2=rb2Δr=rb−ra=rbWith these, the EN definition (263) acquires the atomic representation through the central field potential
|V(r¯)|=Zeffr¯where Zeff stands for the Slater effective atomic number specific for the multi-electronic atoms, being derived from the standard atomic number Z by subtracting the shielding effects of the inner electrons [79]. Nevertheless, worth mentioning that the usual negative sign in attractive potentials was formally abolished because:
▪ we retain the positive values of electronegativity (263) since EN is evaluated as a stability measure of such nuclear-electronic system;
▪ the sign is in accordance with the electric field orientation that drives the sense of the electronic conditional probability of the imaginary evolution amplitude evaluated from the center of atom (ra = 0) to the current valence shell radius (rb).
Therefore, the electronegativity can be seen also as power of holding electrons in the valence shell reciprocal to that exercised upon them from the center of atom. This way, the present EN definition and equation stand for the reconciliation of the two opposite phenomena acting upon the valence electrons: attraction to nucleus and repulsion from the other atomic inner electrons.
Next, within the Bohr description of the electrons moving in a central potential [80], while adopting the atomic units, m = ħ = e2 / 4πε0 = 1, further atomic dependency is acquired by the Bohr-Slater quantifications
rb=n2Zeffτ=2πn3Zeff2in terms of the Slater charge Zeff and the principal quantum numbers of the atomic shell n. These relations are consistent with the above stipulated driven sense of the electronic evolution amplitude (or waves across the orbits), since for the center of the atom they specialize to ra = 0 and τ = 0 in the absence of any orbit (n = 0). Yet, this is the Bohr semiclassical level of the present approach.
However, for keeping the analyticity of the present approach, the computation of the integral (263) with the replacements (265)–(268) may use the saddle-point recipe, very well accommodated for the present semiclassical context; thus we implement the approximation rule [81]
I=∫g(rb)exp[f(rb)]drb≅g(rb0)exp[f(rb0)]−2πf″(rb0)with
rb0=2π2/3n2Zeffcorresponding to the valence shell saddle radius expressed out by the optimization condition
f′(rb0)=0.
According with the exposed strategy, one finds the fourth order semiclassical expansions for electronegativity is given [72]
χ[IV]SC(n,Zeff)≅χ0SC+χ1SC+χ2SC+χ3SC+χ4SCwith the components
χ0SC=Zeff2exp(−3nπ1/3)23π2/3n2χ1SC=−Zeff2[12+22π1/3n+6π2/3n2+5Zeff]exp(−3nπ1/3)303πn3χ2SC=Zeff2(9+2π1/3n)(6+5π1/3n)exp(−3nπ1/3)903π2/3n2χ3SC=−Zeff2(11+5π1/3n)exp(−3nπ1/3)453π1/3nχ4SC=Zeff2exp(−3nπ1/3)363while for the chemical hardness evaluation the relationship (264) is employed to yield the expansion
η[IV]SC(n,Zeff)=[∂∂ZχSC(n,Z)]z→zeff≅η0SC+η1SC+η2SC+η3SC+η4SCwith the components
η0SC=Zeffexp(−3nπ1/3)3π2/3n2η1SC=−Zeff[24+44π1/3n+12π2/3n2+15Zeff]exp(−3nπ1/3)303πn3η2SC=Zeff(9+2π1/3n)(6+5π1/3n)exp(−3nπ1/3)453π2/3n2η3SC=−Zeff2(11+5π1/3n)exp(−3nπ1/3)453π1/3nη4SC=Zeffexp(−3nπ1/3)183
Aiming to unfold the electronegativity and chemical hardness atomic scales, through applying the Equations (271) and (272), the input parameters of the Table 1, along the calibration step between the theoretically and experimentally values of the electronegativity and chemical hardness for the H atom, 7.18 eV and 6.45 eV, are employed, respectively; they provide the numerically energetic pre-factors
χSC:27.21×251.14ηSC:27.21×137.576
The numerical fourth order semiclassical electronegativity and chemical hardness atomic scales are reported in Table 2 and represented in Figure 3, where the comparison with the finite-difference counterparts of Table 1 was also emphasized.
The striking difference in terms of orders of magnitudes observed between elements down groups is the main characteristic of the actual atomic scales of electronegativity and chemical hardness; however, due to the fact the actual definition of electronegativity and chemical hardness reflects the holding power with which the whole atom attracts valence electrons to its center - this is not a surprising behavior.
It is therefore natural to observe that as the atom is richer in core electrons down groups lesser is the attractive force on the outer electrons from the center of the atom. In this regard, the actual scales mirror the atomic stability of the valence shell at the best.
Nevertheless, a better regularization of their increasing trend along periods it is observed in Figure 3 for both semiclassical electronegativity and chemical hardness fourth order scales, a feature more apparent for the actual chemical hardness scale. Moreover, since the chemical hardness controls the secondary order effects throughout its definition as the derivative of the electronegativity a phenomenological rule would demand to have lower values than that of the associated electronegativity.
However, this rule, while being not always obeyed for the finite difference ηFD values (e.g., He, Ne, Ar, Kr, Xe) is well satisfied with the present semiclassical ones
η[IV]SC as compared with their counterpart electronegativities, χFD and
χ[IV]SC, respectively.
Going to electronegativity discussion, the present
χ[IV]SC values seem to respect almost all empirical criteria for acceptability [83]:
the atoms N, O, F, Ne, and He have the highest electronegativities among the main groups;
the electronegativity of N is by far greater than that of Cl - a situation that is not met in the finite-difference approach;
the Silicium rule demanding that most metals to have EN values less than or equal to that of Si, is as well widely satisfied;
the metalloid band (B, Si, Ge, As, Sb, Te) clearly separates the metals by nonmetals’ EN values;
along periods the highest EN values belong to the noble elements – a rule not fulfilled by the couples (Cl, Ar), (Br, Kr), and (I, Xe) within the finite difference representation, see Table 1;
the recorded electronegativity values of the chalcogens (O, S, Se, Te) reveal great distinction between the chemistry of oxygen and the rest elements of VIA group;
the transitional metals are grouped in a distinct contracted region of EN values – this way closely emphasizing on the d-orbitals effects, a criteria almost not fulfilled by the finite-difference scheme, see Table 1 and the Figure 3.
Finally, we have to point out that the systematic decrease of orders of magnitude of electronegativity and hardness semiclassical scales of Table 2 and Figure 3 has a fundamental consequence, namely it stands as the computational proof that the electronegativity and hardness behave like pure quantum/structural indices. As such, they are not manifesting with the same intensity among all elements of the Periodic Table by having values that tend to considerably diminish as the frontier electrons are farer and feel less and less the quantum influence (potential and force) of the nucleus and of the core electrons. These results are in accordance with the electronic localization principles in an atom [3,84].
Effective Classical Path Integral of Evolution AmplitudeEffective Classical Partition Function
As previously shown, see Equations (121c) and (208), for instance, considering the path integral propagator that underlies the canonical density in the quantum statistical algorithm, see Equations (87)–(89), accounts for the quantum effects (fluctuations) induced on single particle paths by the presence of an external potential, while being analytically computed by averaging these over all possible configurations. Yet, one could observe that for periodic paths, i.e., when the final and initial space-points coincide
xa=xbthe particle travels in very short time not far away from the initial position and then is back on the initial point; such picture has the physical measurable consequence a particle is observed on the initial point, i.e., it is found on a stationary state/orbit, while the quantum fluctuations are oscillating around the equilibrium (initial = final) space-point. Even clearer, the situation corresponds to the classical picture in which a particle behaves, being accommodated in an equilibrium state/stationary orbit under external potential influence. This means that the external influence itself is observable in (initial = final) concerned/measured state, thus being no longer a path parameterized function, but a constant:
V(x(τ),τ)→V(xa)
Therefore, the associated periodic propagator (density matrix) becomes
(xaℏβ;xa0)=e−βV(xa)∫x(0)=xax(ℏβ)=xaD′x(τ)exp{−1ℏ∫0ℏβ[m2x˙2(τ)]dτ}︸FREEMOTIONPROPAGATOR=e−βV(xa)m2πβℏ2where the recognized path integral of free motion was solved by plugging into its quantum mechanical solution (171) the present conditions (274) and (196b), for accounting of the path periodicity and quantum statistics, respectively.
At the same time there is clear that the periodic path condition (274) is not arbitrarily but a compulsory step since characteristic in passing from density matrix to partition function and then to the real (measurable or workable) canonical and N-particle density, according with the density matrix algorithm (87)–(89). Therefore, the resulting partition function built from the un-normalized canonical density (276) assumes the simple form
Zcl=∫(xaℏβ;xa0)dxa=m2πβℏ2∫e−βV(xa)dxawhile being susceptible of universal reliability if not limited by the degree the periodicity between the final and initial space-point is achieved through condition (274). However, looking to free motion path integral solution (171) we see that the classical observation is readily valid for the coordinate departure not exceeding the critical value
xb−xa=Δxcl=ℏβmin which case the exponential limit
exp{−m2ℏ2β(xb−xa)2}→xb−xa=Δxcle−1/2=0.607is approximated with unity in expression (276), thus with an error of 40% at the maximum displacement of (278) value; as the classical displacement (278) tends to zero as the expressions (276) and (277) become more accurate. Following the Feynman standard example, for a crystal with atoms of typical atomic mass (A) about 20, at room temperature, the classical limit of displacement (278) gives about 0.1 Å; this is the maximum displacement of those atoms around their equilibrium position in the lattice when the thermodynamic properties of the solid can be evaluated through considering the classical form of partition function (277). Just in passing worth noting that the partition function (277) is called “classical” despite carrying the exponential pre-factor with the quantum Planck constant since the configuration integral ∫exp(– βV) was historically anticipated and worked out by Boltzmann, in the pre-quantum era with a non-specified multiplying constant, known today as the inverse of the so called thermal lengthλth=2πβℏ2m
With these considerations there appears as natural the generalization of the classical partition form (277) into the more comprehensive one known as the effective classical partition function [85–90]
Zeff−cl=∫−∞+∞dx02πℏ2β/mexp[−βVeff−cl(x0)]with the integration variable defined as the thermal average of the periodic quantum paths
x0≡x¯=1ℏβ∫0ℏβx(τ)dτsometimes called as the Feynman centroid, while the notation is to be right bellow justified.
Moreover, the search for the best approximation of effective-classical partition function (280) will be conducted as such the quantum fluctuations be not dependent on the classical displacement (278), abstracted from the free motion, but being driven by the quantum harmonic oscillations – through they constitute a generalization of the free motion itself, see for instance the equivalence of classical paths or propagators of free with harmonic motion in the zero-frequency limit, see the Section 3.3.3.
However, the periodicity condition (274) for paths is to be maintained and properly implemented in approximating the effective-classical partition function (281) being, nevertheless, closely and powerfully related with the quantum beloved concept of stationary orbits defined/described by periodic quantum waves/paths. This way, the effective-classical path integral approach appears as the true quantum justification of the quantum atom and of the quantum stabilization of matter in general, providing reliable results without involving observables or operators relaying on special quantum postulates other than the variational principles – with universal (classical or quantum) value.
Periodic Path IntegralsMatsubara Frequencies and the Quantum Periodic Paths
As always done when a new type of path integral is under consideration the reconsideration of the quantum paths, and in fact the quantum fluctuations, is undertaken so that facilitating the best way for solving it. Yet, this time due to the periodicity condition of paths the propagator is hidden by the associated partition function. Therefore, the optimum approximation for the effective classical potential in (281) will provide the periodic evolution amplitude as well, i.e., the un-normalized density, which by normalization with partition function will lead with the searched canonical density counterpart.
Going to characterize the periodic paths, they will be seen as the Fourier series [34,39,41,91,92]
x(τ)=∑m=−∞+∞xmexp(iωmτ)in terms of the so called Matsubara frequencies ωm; they are explicitly found through specializing the condition (274) into the actual statistical one, see Figure 4xa=x(0)=x(ℏβ)=xbresulting in the equality
1=exp(iωmℏβ)with the solution
ωm=2πℏβm,m∈Zwhich certifies the quantization of the paths (282a).
Moreover, under the condition the quantum paths (282a) are real
x*(τ)=x(τ)the equivalent expanded form with the conjugated path
x*(τ)=∑m=−∞+∞xm*exp(−iωmτ)=∑m=−∞+∞xmexp(iωmτ)yields for the coefficients of the periodical paths the relationship
xm*=x−m=xm
With this, the quantified form of periodic path frequencies, Equation (283), allows separating the paths (282a) into the constant and complex conjugated oscillating contributions
x(τ)=x0+∑m=1+∞xmexp(iωmτ)+∑m=−1−∞xmexp(iωmτ)=x0+∑m=1+∞xmexp(iωmτ)+∑m=1+∞x−mexp(−iωmτ)=x0+∑m=1+∞xmexp(iωmτ)+c.c.with the 0th terms viewed more than the “zero-oscillating” or free motion path but the thermal averaged path over entire quantum paths (282a)
1ℏβ∫0ℏβx(τ)dτ=1ℏβ∫0ℏβ[x0+∑m=1+∞xmexp(iωmτ)+c.c.]dτ=x0+∑m=1+∞xm1ℏβ∫0ℏβexp(iωmτ)dτ+∑m=1+∞x−m1ℏβ∫0ℏβexp(−iωmτ)dτ=x0+1ℏβ∑m=1+∞xm∫0ℏβ[exp(iωmτ)+exp(−iωmτ)]dτ=x0+2ℏβ∑m=1+∞xm∫0ℏβcos(ωmτ)dτ=x0+2ℏβ∑m=1+∞xmωmsin(ωmℏβ)=x0+2ℏβ∑m=1+∞xmωmsin(2πm)m∈Z︸0︸0thus resulting in the Feynman centroid formula (281b).
However, beside revealing the integration variable of the classical partition function (281a) as being of averaged nature, the result (282e) emphasizes on the actual periodic path decomposition (282d) integral featuring another level for parameterization of quantum paths that goes beyond characterizing them as quantum fluctuations around classical motion; they are here constructed as periodic oscillations (back and forth – due to their complex form, in analogy with conjugated plane waves traveling in opposite directions) around the averaged path value (interpreted as thermic average, or, more plastic, as centroid of the quantum fluctuations themselves). Therefore, with such path parameterization perspective the present level seems involving quite complex quantum phenomenology to be further enriched in the sections to follow.
Matsubara Harmonic Partition Function
The quantum path decomposition (282d) imposes the factorization of the Feynman path integral measure (120) accordingly
∫x(0)=x(ℏβ)D″x(τ)≡(C0∫−∞+∞dx0)[∏m=1∞Cm∫−∞+∞dRe(xm)∫−∞+∞dIm(xm)]with the integration constants C0 and Cm to be determined from identifying the known partition function of the harmonic oscillator, Equation (192), with the path integral representation of the counterpart partition function written by the measure (287) and paths (282)
ZΩ=∫x(0)=x(ℏβ)D″x(τ)exp{−1ℏ∫0ℏβdτ[m2x˙2(τ)+m2Ω2x2(τ)]}=(C0∫−∞+∞dx0)[∏m=1∞Cm∫−∞+∞dRe(xm)∫−∞+∞dIm(xm)]exp{−1ℏ∫0ℏβdτ[m2x˙2(τ)+m2Ω2x2(τ)]}=!12sinh(ℏβΩ/2)
To this end, let’s begin with the computation of the kinetic term appearing under the integral (288a)
m2∫0ℏβdτx˙2(τ)=m2∫0ℏβdτ[∑m=−∞+∞ixmωmexp(iωmτ)][∑m′=−∞+∞ixm′ωm′exp(iωm′τ)]=−m2∑m=−∞+∞∑m′=−∞+∞xmxm′ωmωm′∫0ℏβdτexp[i(ωm+ωm′)τ]=−m2ℏβ∑m=−∞+∞∑m′=−∞+∞xmxm′ωmωm′δm+m′,0=−m2ℏβ∑m=−∞+∞xmx−mωmω−m=ω−m=−ωmm2ℏβ∑m=−∞+∞ωm2|xm|2=mℏβ∑m=1+∞ωm2[(Rexm)2(Imxm)2]while for the harmonic contribution we similarly get
m2Ω2∫0ℏβdτx2(τ)=m2Ω2∫0ℏβdτ[∑m=−∞+∞xmexp(iωmτ)][∑m′=−8+∞xm′exp(iωm′τ)]=m2Ω2∑m=−∞+∞∑m′=−∞+∞xmxm′∫0ℏβdτexp[i(ωm+ωm′)τ]=m2Ω2∑m=−∞+∞∑m′=−∞+∞xmxm′ℏβδm+m′,0=m2Ω2ℏβ∑m=−∞+∞xmx−m=m2Ω2ℏβ∑m=−∞+∞xmxm*=m2Ω2ℏβ{x02+2∑m=1+∞[(Rexm)2+(Imxm)2]}all together be combined in the partition function (288a)
ZΩ=(C0∫−∞+∞dx0)[∏m=1∞Cm∫−∞+∞dRe(xm)∫−∞+∞dIm(xm)]×exp{−mβ∑m=1+∞ωm2[(Rexm)2+(Imxm)2]−m2Ω2βx02−mΩ2β∑m=1+∞[(Rexm)2+(Imxm)2]}=[C0∫−∞+∞dx0exp(−m2Ω2βx02)]∏m=1∞Cm{∫−∞+∞dRe(xm)exp[−mβ∑m=1+∞(ωm2+Ω2)(Rexm)2]}2=C02πmβΩ2∏m=1∞Cmπmβ(ωm2+Ω2)
Next, with the frequency choice
Cm=Cωm2the partition function (290) further resumes as
ZΩ=C01Ω2πmβπmβCf(Ω)noting the newly introduced function
f(Ω)=∏m=1∞ωm2ωm2+Ω2like a series of Matsubara frequencies. Therefore, it remains that in order the partition function (292) be solved the product series (293a) has to be evaluated; this new issue may be accomplished through three more transformations, namely, by firstly rewriting the series (293a) as
f(Ω)=exp[g(Ω)]
With:
g(Ω)=∑m=1∞lnωm2ωm2+Ω2followed by considering its derivative
g′(Ω)=∑m=1∞ωm2+Ω2ωm2−ωm2(ωm2+Ω2)22Ω=−2ΩR(Ω)leaving with the final evaluation of the Riemann generalized series
R(Ω)=∑m=1∞1Ω2+ωm2
Going back, once calculated, the Riemann series (296) is replaced in (295) which, at its turn, is employed for the integral evaluation
g(Ω)=g(0)+∫0ΩdΩ˜g′(Ω˜)for ending with unfolding of the searched function (293). These steps will be systematically exposed in next sections.
The Generalized Riemann’ Series
Computation of the generalized Riemann series (296) requires few intermediate operations:
○ Writing it under the form
R(Ω)=12[F(Ω)−1Ω2]in terms of the extended series
F(Ω)=∑m=−∞∞1Ω2+ωm2=ℏ2β24π2∑m=−∞∞1α2+m2,α≡ℏ2β2Ω24π2
○ Applying the Poisson formula, see Appendix (A8), on series (298b)
F(Ω)=ℏ2β24π2∑n=−∞∞[∫−∞+∞dq1q2+α2exp(−i2πqn)]
○ Computing the integral under the sum of (299) by the complex integration, according with the contours of integration identified in Figure 5 around the poles q = ±iα throughout applying the residues’ theorem
∮C(z0)f(z)dz=2πiRezz=z0f(z)=2πilimz−>z0(z−z0)f(z)while summing upon the convergent cases
n≥0⇒Imq<0⇒contourIIn<0⇒Imq>0⇒contourIarisen from the observation that
|exp[−i2π(Req+iImq)n]|=exp(2πnImq)Note that the contour (I) is considered completed with trigonometric positive direction, while for the (II) contour the anti-trigonometric sense results, as being equivalent with the minus sign in front of its integral, which, explicitly gives
∫−∞(II)+∞dq1q2+α2exp(−i2πqn)=(−1)∮C(II)dq1q2+α2exp(−i2πqn)=−2πiRezq=−iaexp(−i2πqn)q2+α2=−2πilimq=−iaα=ℏβΩ2π(q+iα)exp(−i2πqn)(q+iα)(q−iα)=2π2ℏβΩexp(−ℏβΩn)while the other contour integration leads with similar result.
○ Insertion of these integration results in the expression (299) is done by attributing to each contour and integration the (series) summing range according with the constraints of (301a) to successively yield
F(Ω)=ℏ2β24π2[∑n=1∞∫−∞(I)+∞dqexp(i2πqn)q2+α2+∑n=0∞∫−∞(II)+∞dqexp(−i2πqn)q2+α2]=ℏ2β24π22π2ℏβΩ[∑n=1∞exp(−ℏβΩn)+∑n=0∞exp(−ℏβΩn)]=ℏβ2Ω{2∑n=0∞[exp(−ℏβΩ)]n−1}=ℏβ2Ωcoth(ℏβΩ2)
○ With expression (303) the Riemann series (298a) finally reads as
R(Ω)=∑m=1∞1Ω2+ωn2=ℏ2β24π2∑m=1∞1(ℏβΩ/2)2+m2=14ℏβΩ[coth(ℏβΩ2)−2ℏβΩ]
○ The cross-check with the usual Riemann series is performed by means of turning the harmonic to free motion picture, as the already consecrated free to harmonic motion interplay; That is to evaluate the limit
limΩ→0R(Ω)=ℏ2β24π2∑m=1∞1m2=limΩ→014ℏβΩ[coth(ℏβΩ2)−2ℏβΩ]within the hyperbolic cotangent function approximation
[coth(Ξ)]Ξ→0≅1Ξ+Ξ3to give the identity
ℏ2β24π2∑m=1∞1m2=14ℏ2β26leaving with the classical Riemann series limit
∑m=1∞1m2=π26
We have now clarified all prerequisites to readily compute the Matsubara harmonic partition function (292), used as a tool to find out the Matsubara normalization of periodic path integrals. This will be addressed in the sequel.
Periodic Path Integral Measure
The Matsubara harmonic partition function algorithm (292)–(297) may be now unfolded successively as:
○ Computing the function (295) by inserting the above Riemann generalized series (304):
g′(Ω)=−2ΩR(Ω)=−ℏβ2[coth(ℏβΩ2)−2ℏβΩ]
○ Evaluating the function (294) by the aid of (297) rule through considering the variable change z = ħβΩ/ 2 in (309)
g(Ω)=−∫0ℏβΩ/2dz(cothz−1z)=−[ln(sinhz)−lnz]0ℏβΩ/2=−[ln(sinhzz)]0ℏβΩ/2=ln[ℏβΩ2sinh(ℏβΩ/2)]
○ Obtaining the function (293a) with the help of (293b) and (310)
f(Ω)=∏m=1∞ωm2ωm2+Ω2=exp[g(Ω)]=ℏβΩ2sinh(ℏβΩ/2)
○ Releasing the Matsubara partition function for the harmonic motion by replacing function (311) into expression (292)
ZΩ=C0C2πmβπℏm12sinh(ℏβΩ/2)
○ Comparing the form (312) with the consecrated results (192) or (288b), thus getting the condition
1=C0C2πmβπℏm
○ Choosing for the Feynman centroid normalization factor the inverse of the thermal length (280)
C0=1λth=12πℏ2β/m
○ Plugging expression (314) in (313) to yield the constant
C=βmπand then by considering it into the relation (291) to get the Matsubara constants
Cm=Cωm2=βm0ωm2π
○ Replacing the constants (314) and (315b) in (287) to provide the normalized measure of the periodic integrals in terms of the Matsubara quantum frequencies (283)
Note that the measure given in (316) is rather universal for periodic paths, while the involvement of the harmonic oscillator was only a tool (and always an inspiring exercise) for determining it since the complete quantum and statistical solution at hand. As such, once more, the harmonic motion proofs its versatile properties respecting the fluctuation over – or perturbation of – the free motion by modeling the quantum displacements from classical equilibrium or observed path.
Feynman-Kleinert Variational FormalismFeynman-Kleinert Partition Function
Being equipped with the periodic path integral technique we can present one of the most efficient ways for approximate the effective-classical partition function (281a); it starts with the general path integral
Z=∫x(0)=x(ℏβ)D″x(τ)exp{−1ℏS+[x,x˙,τ]}=∫x(0)=x(ℏβ)D″x(τ)exp{−1ℏ∫0ℏβdτ[mx˙2(τ)2+V(x(τ))]}=(∫−∞+∞dx02πℏ2β/m)[∏m=1∞∫−∞+∞∫−∞+∞dRexmdImxmπ/(mβωm2)]exp[−βm∑m=1∞ωm2|xm|2−1ℏ∫0ℏβdτV(x(τ))]
Since the analytical solution for expression (317) is hard to be conceived for an unspecified potential form, it may be eventually reformulated in a workable from by involving another partition function, the so called Feynman-Kleinert partition function ZFKZ=∫x(0)=x(ℏβ)D″x(τ)exp{−1ℏSFK+[x,x˙,τ]}exp{−1ℏ(S+[x,x˙,τ]−SFK+[x,x˙,τ])}=ZFK〈exp{−1ℏ(S+[x,x˙,τ]−SFK+[x,x˙,τ])}〉FKand its special average recipe
〈O[x]〉FK=ZFK−1∫x(0)=x(ℏβ)D″x(τ)O[x]exp{−1ℏSFK+[x,x˙,τ]}
In Equations (318a) and (318b) the Feynman-Kleinert partition function writes as [85]
ZFK=∫x(0)=x(ℏβ)D″x(τ)exp{−1ℏSFK+[x,x˙,τ]}with the working action ansatz
SFK+[x,x˙,τ]=∫0ℏβdτ[mx˙2(τ)2+m2Ω2(x0)(x(τ)−x0)2+LFK(x0)]
The Feynman-Kleinert partition function is constructed as such, unlike the general partition function (317), to explicitly account for the path fluctuations around the Feynman centroid (281b) through the term (x(τ) − x0)2, driven harmonically by the frequency Ω2 (x0), with a role in optimizing the quantum fluctuations in order state equilibrium be achieved. The supplementary Feynman-Kleinert perturbation function LFK (x0) assures the global optimization for the action, and implicitly for the Feynman-Kleinert partition function, so approaching at the best the exact partition function (317) and its associate total ground state energy of the system given by the free energy
F=−β−1lnZ
In fact, the Feynman-Kleinert action (319b) is to be involved in two-fold optimization algorithm for providing the best approximation of the partition function (317). This will favor a close analogy with the double search for electronic density, in density functional theory (DFT), as will be discussed later.
Yet, the Feynman-Kleinert partition function is to be unfolded within the actual periodic path integral representation
ZFK=∫x(0)=x(ℏβ)D″x(τ)exp{−1ℏ∫0ℏβdτ[mx˙2(τ)2+m2Ω2(x0)(x(τ)−x0)2+LFK(x0)]}=(∫−∞+∞dx0exp[−βLFK(x0)]2πℏ2β/m)[∏m=1∞∫−∞+∞∫−∞+∞dRexmdImxmπ/(mβωm2)]exp{−mβ∑m=1∞[ωm2+Ω2(x0)]|xm|2}︸f(Ω)seeeqs.(292)&(311)=∫−∞+∞dx02πℏ2β/mℏβΩ(x0)/2sinh(ℏβΩ(x0)/2)exp[−βLFK(x0)]while being resumed under the effective-classical form (281a)
ZFK=∫−∞+∞dx02πℏ2β/mexp[−βWFK(x0)]by means of the Feynman-Kleinert potential [85]
WFK(x0)=1βln[sinh(ℏβΩ(x0)/2)ℏβΩ(x0)/2]+LFK(x0)to be optimized respecting its harmonic frequency (for equilibrium optimization) and for ground state perturbation (optimization) in what follows.
Feynman-Kleinert Optimum Potential
The optimization of the Feynman-Kleinert partition function (322) is performed employing the Jensen-Peierls inequality
〈exp[O]〉≥exp[〈O〉]whose the phenomenological proof is given in the Figure 6, on the partition function relationship (318a) leading to the lower bounded partition function
Z≥ZFKexp{−1ℏ〈S+[x,x˙,τ]−SFK+[x,x˙,τ]〉FK}or, by calling the Equation (320), to the higher bounded free energy
F≤FFK+1ℏβ〈S+[x,x˙,τ]−SFK+[x,x˙,τ]〉FK
When rewritten the last inequality with the help of Euclidian actions for general partition function and the Feynman-Kleinert specialization, Equations (317) and (319), respectively, one notes the cancellation of the kinetic (free motion) terms, while the resulting expression
F≤FFK+1ℏβ∫0ℏβdτ〈V(x(τ))−m2Ω2(x0)(x(τ)−x0)2−LFK(x0)〉FKprovides the searched variational architecture
δF=0⇒〈V(x(τ))−m2Ω2(x0)(x(τ)−x0)2−LFK(x0)〉FK=0where all involved terms combine the external, perturbation and quantum fluctuation influences.
Having the variational problem formulated it remains to individually compute the terms appearing in the Feynman-Kleinert average (328), by using the associate definition (318b) with the action (319b).
Going to evaluate the most general term containing the external potential average, we have in the first instance its periodic path integral representation
〈V(x(τ))〉FK=ZFK−1∫x(0)=x(ℏβ)D″x(τ)V(x(τ))exp{−1ℏSFK+[x,x˙,τ]}=ZFK−1(∫−∞+∞dx0exp[−βlFK(x0)]2πℏ2β/m)[∏m=1∞∫−∞+∞∫−∞+∞dRexmdImxmπ/(mβωm2)]exp{−mβ∑m=1∞[ωm2+Ω2+(x0)]xm|2}×∫−∞+∞dk2πV(k)exp{ik[x0+∑n=1∞xnexp(−iωnτ)+c.c.]}=ZFK−1∫−∞+∞dx0exp[−βLFK(x0)]2πℏ2β/m∫−∞+∞dk2πV(k)exp(ikx0)×[∏m=1∞∫−∞+∞∫−∞+∞dRexmdImxmπ/(mβωm2)]exp{−mβ∑m=1∞[ωm2+Ω2(x0)]xm|2+ik[∑n=1∞xnexp(−iωnτ)+c.c.]}=ZFK−1∫−∞+∞dx0exp[−βLFK(x0)]2πℏ2β/m∫−∞+∞dk2πV(k)exp[ikx0−12a2(x0)k2]×[∏m=1∞∫−∞+∞∫−∞+∞dRexmdImxmπ/(mβωm2)]exp{−mβ∑m=1∞[ωm2+Ω2(x0)][(Rexm−ik1/(mβ)ωm2+Ω2(x0)cosωmτ)2+(Imxm−ik1/(mβ)ωm2+Ω2(x0)sinωmτ)2]}︸f(Ω)seeeqs.(290),(292),(293a),(311)&(315b)=ZFK−1∫−∞+∞dx0exp[−βLFK(x0)]2πℏ2β/mℏβΩ(x0)/2sinh(ℏβΩ(x0)/2)∫−∞+∞dk2πV(k)exp[ikx0−12a2(x0)k2]where the Fourier k- (wave vector) representation was implemented for external potential so that the quantum path to explicitly appear in evaluation; this technique had helped for performing the quadratic completion of paths in the view of harmonic-like integration of type (290) with the result (311); nevertheless, in course of these operations the new quantity was introduced, namely the fluctuation width
a2(x0)=21mβ∑m=1∞1ωm2+Ω2(x0)which can be immediately recognized as directly related with generalized Riemann series (304), thus having the form [85]
a2(x0)=21mβR(Ω)=1mβΩ2[ℏβΩ2coth(ℏβΩ2)−1]
Expression (329) can be even more simplified when solving out the k-integral by back considering the Fourier transformation for the potential and then proceeding with the quadratic completion toward the Poisson standard integration
Va2(x0)(x0)=∫−∞+∞dk2πV(k)exp[ikx0−12a2(x0)k2]=∫−∞+∞dk2π∫−∞+∞dxV(x)exp(−ikx)exp[ikx0−12a2(x0)k2]=12π∫−∞+∞dxV(x)exp[−(x−x0)22a2(x0)]∫−∞+∞dkexp[(x−x02a(x0)−ia(x0)2k)2]=12π∫−∞+∞dxV(x)exp[−(x−x0)22a2(x0)]∫−∞+∞dk′exp[−a2(x0)2k′2]=12πa2(x0)∫−∞+∞dxV(x)exp[−(x−x0)22a2(x0)]≡〈V(x)〉a2(x0)
The potential (331) is known as the smeared out potential and has a major role in explaining the quantum stabilization of matter, as will be largely discussed in the next section. For the moment it is regarded jus as the integral transformation of the original applied potential by convolution with a Gaussian packet with the width a2(x0) that accounts for the existing quantum fluctuation in the system.
Nevertheless, Equation (331) leaves the Feynman-Kleinert average of external potential with the result [85]
〈V(x(τ))〉FK=ZFK−1∫−∞+∞dx02πℏ2β/m〈V(x)〉a2(x0)ℏβΩ(x0)/2sinh(ℏβΩ(x0)/2)exp[−βLFK(x0)]
For the rest of the averaged terms in (328) the evaluations are considerably easier since for each of them we have only to compute their smeared out version (331), with the yield for the trial harmonic
〈m2Ω2(x0)(x−x0)2〉a2(x0)=m2Ω2(x0)12πa2(x0)∫−∞+∞dx(x−x0)2exp[−(x−x0)22a2(x0)]=m2Ω2(x0)a2(x0)and respectively for the Feynman-Kleinert trial function
〈LFK(x0)〉a2(x0)=LFK(x0)12πa2(x0)∫−∞+∞dxexp[−(x−x0)22a2(x0)]=LFK(x0)
When replaced in average form (332) the forms (333) and (334) cumulate with the smeared out potential (331) in the final Feynman-Kleinert average equation (328), now featuring the form
0=∫−∞+∞dx02πℏ2β/m{Va2(x0)(x0)−m2Ω2(x0)a2(x0)−LFK(x0)}ℏβΩ(x0)/2sinh(ℏβΩ(x0)/2)exp[−βLFK(x0)]from where the first stage of variational algorithm is fulfilled by the obvious choice
LFK(x0)=Va2(x0)(x0)−m2Ω2(x0)a2(x0)
With Equation (336) the Feynman-Kleinert potential (323) now displays as [85]
WFK(x0)=1βln[sinh(ℏβΩ(x0)/2)ℏβΩ(x0)/2]+Va2(x0)(x0)−m2Ω2(x0)a2(x0)
There remains only to finally optimize the explicit potential (337) for the harmonic (trial) frequency assuring therefore the equilibrium of the gained lowest approximation of the ground state for the concerned system. This is simply achieved through the chain derivative
0=dWFK(x0)dΩ2(x0)=∂WFK(x0)∂Ω2(x0)+∂WFK(x0)∂a2(x0)∂a2(x0)∂Ω2(x0)seeing that also the fluctuation width (330) depends on harmonic frequency. Moreover, due to the derivative equivalence 2(∂/ ∂Ω2)Ξ = (1/ Ω)(∂/ ∂Ω)Ξ the first term in (338) is arranged to emphasize on its vanishing nature when recalling the Equation (337) and the fluctuation width (330b)
∂WFK(x0)∂Ω2(x0)=m2{1mβΩ2[ℏβΩ2coth(ℏβΩ2)−1]−a2(x0)}=0
This way, from Equation (338) it remains only the simple condition
∂WFK(x0)∂a2(x0)=0that provides for (337) the optimum (stabilization) frequency of quantum fluctuation
Ω2(x0)=2m∂Va2(x0)(x0)∂a2(x0)
Nevertheless, through observing the huge role both the smeared out potential (331) and the fluctuation width (330) play in deriving the approximated equilibrium ground state they deserve be further analyzed and commented in relation with matter stability.
Quantum Smeared Effects and the Stability of Matter
The intriguing role the smeared potential in special and the smearing effect in general play in optimization of the total energy and partition function of a quantum system opens the possibility analyzing the “smearing” phenomenon of the quantum fluctuation in a more fundamental way.
I. Firstly, it was noted that the smearing potential (331) appears as a Gaussian convolution of the applied potential, although modeling the evolution of a wave-packet under that potential influence; in other terms, it appears the fundamental question whether the Gaussian and wave function “kernels” behave in similar way throughout the smearing effect of quantum fluctuations; analytically, one likes to see whether the next smearing average equality readily holds
〈exp(−ikx)〉a2(x0)=?〈exp(−k2x2)〉a2(x0)
In order to check (342) one separately computes each of its sides by the aid of k-form of (331) and successively gets the smearing average for the wave-function
〈exp(−ikm)〉a2(x0)=∫−∞+∞dk2πexp[−ikx+ikx0−12a2(x0)k2]=∫−∞+∞dk2πexp[−ik(x−x0)−12a2(x0)k2]=exp[−(x−x0)22a2(x0)]∫−∞+∞dk2πexp{−a2(x0)2[k+ix−x0a2(x0)]2}=12πexp[−(x−x0)22a2(x0)]∫−∞+∞dk′exp{−a2(x0)2k′2}=12πa2(x0)exp[−(x−x0)22a2(x0)]and for the Gaussian packet
〈exp(−k2x2)〉a2(x0)=∫−∞+∞dk2πexp[−k2x2+ikx0−12a2(x0)k2]=∫−∞+∞dk2πexp[−k2(x2+a2(x0)2)+ikx0]=exp[−x024(x2+a2(x0)/2)]∫−∞+∞dk2πexp{−(x2+a2(x0)/2)[k−ix02(x2+a2(x0)/2)]2}=12πexp[−x024(x2+a2(x0)/2)]∫−∞+∞dk′exp{−(x2+a2(x0)/2)k′2}=12π[2x2+a2(x0)]exp[−x022(2x2+a2(x0))]
Now, for closely comparison of the expressions (343a) and (343b) the most elegant way is to make once more recourse to the smearing procedure, this time referring both to the entire paths and Feynman centroid; to this end, the previous result (333) is here used in the variant:
〈(x−x0)2〉a2=a2
It allows the additional similar relationships
〈x2〉a2=〈x02〉a2=a22
Note that the equality (344b) is due to the symmetry of the smearing average formula (331) at the interchange x ↔ x0, while the mixed term of (344a) expansion vanishes, 〈xx0〈 a2(x0) = 0, in any path representation. With these rules, one can reconsider Equations (343a) and (243b) by performing the formal equivalences
(x−x0)2≈a2x2≈a22x02≈a22yielding with
〈exp(−ikx)〉a2≈12πa2exp[−12]〈exp(−k2x2)〉a2≈14πa2exp[−18]=〈exp(−ikx)〉a2exp(3/8)2Since the difference between these expressions is numerically proportionally with the factor
exp(3/8)2≅1.029they can be considered as identical in quantum smearing effects and Equation (342) as valid.
Yet, the quantum identity between the plane-wave and Gaussian packet has profound quantum implication, while revealing for instance the de Broglie – Born identity in Gaussian normalization of the de Broglie moving wave-packet. It may express as well the observational Gaussian character of the wave-function evolution in Hilbert space. Finally, and very important, it leads with explanation of the Bohr first postulate, i.e., it is able to explain the stationary wave on orbits under singular (Coulombic) potential thus explaining the matter stabilization on rigorous quantum base, rather than to admit it by the power of a postulate. This is to be proved next [39,85].
II. Let’s consider a quantum system evolving under the influence of the Yukawa potential, as a generalization of the Coulomb interaction, available also for the sub-nuclear world
VYuk(r)=Arexp(−αr)with r = x −