This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

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.

In modern conceptual and computational chemistry the Density Functional Theory (DFT) [_{i}_{i}_{i}_{i}_{i}_{=
1,N} and {_{i}_{i=
1,N}, respectively, in constructing the basic functional (integral) for the charge conservation
_{ee}

Yet, although accessible experimentally [^{(1)} (_{1}; _{1}) provides by integration the total number of electrons in the same way as the observable density does [^{(1)} (_{1}; _{1})

Moreover, within the thermal-temporal (quantum mechanical-to quantum statistical) Wick transformation
_{B}

The competition between the variation of density functional of energy (3) and the density itself (12),

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
_{a}^{(1)} (_{b}_{a}_{a}

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 [

Given a spectral representation {|_{n}_{∈N} for a set of quantum mono-electronic states
_{k}

Similarly, when rewriting the global average as

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 [

Next, in the case the concerned quantum states are

Moreover, in these eigen-conditions, the operatorial average further reads from

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

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

Yet, the anti-symmetric restriction the _{α}

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

○ Normalization

○ Recursion

○ First order Löwdin reduction

where the first order density matrix casts as abstracted from general definition

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 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

It is obvious that although the second order reduced matrix has appeared, its general form

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

○ The idempotency

○ The normal additivity, see

○ Kernel multiplicity

○ Many-body normalization

Remarkably, the last two identities may serve as the constraints when minimizing the above Hamiltonian average, here appropriately rewritten employing _{1}) quantity producing the actual so called

Performing the functional derivative respecting the Fock-Dirac electron density in (54) one gets the equivalent expression
_{1} – _{1}), with

Giving the idempotency property of

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

Lastly, it should be noted that all above properties may be rewritten since considering the

For a quantum system obeying the _{k}

This is a very interesting and important result motivating the quantum statistical approach in determining the density of states since it corresponds to the

The recognized importance of partition functions in computing the internal energy as the average of the Hamiltonian of the system:
^{–βĤ}

The great importance of density operator of

○ It identifies the evolution operator

○ It produces the so called

○ Fulfills the (short times, higher temperature) so called Markovian limiting condition

In the frame of coordinate representation the Bloch problem,

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

In these conditions, one may firstly write
_{0} (_{0} (_{b}_{a}

Such slicing procedure in solving the Bloch

Now, in the first instance, the new problem (82) has the

The general canonic solution (84) is viewed as the path integral solution for the Bloch

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.

Through reconsidering the slicing of (81) also for the time interval [_{b}_{a}_{b}_{a}

Now, the elementary quantum evolution amplitude (95) is to be evaluated, firstly by reconsidering the eigen-coordinate unitary operator, in the working form

Next, each obtained working energetic contribution is separately evaluated: for kinetic contribution the insertion of the momentum complete eigen-set

For achieving such goal, a more practical form of the Feynman integral may be obtained once the Hamiltonian is implemented as

The remaining quantum evolution amplitude reads as the spatial path integral only

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

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),

Yet, the connection of the path integrals of propagators with the Schrödinger quantum formalism is to be revealed next.

There are two ways for showing the propagator path integral links with Schrödinger equation.

Firstly, by employing one of the above path integral, say that of

Similarly for the second derivative we have

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

Remarkably, besides establishing the link with the Schrödinger picture, _{b}_{b}_{b}_{b}_{a}_{a}

Nevertheless, the path integral formalism is able to provide also

The starting point is the manifested

Yet, we will employ

Next, since noting the square dependence of ^{2} ≅ 0, and were we arranged the exponentials under integrals of Gaussian type (

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,

There are three fundamental properties most useful for path integral calculations [

Firstly, one may combine the two above Schrödinger type bits of information about path integrals: the fact that propagator itself (_{b}_{b}_{a}_{a}_{a}_{b}

Secondly, from the Huygens principle (131) there is abstracted also the limiting delta Dirac-function for a propagator connecting two space events simultaneously

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 _{cl}

Very important, note that the quantum fluctuation vanishes at the end-points of the evolution path since “meeting” with the classical (observed) path, see

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

Actually, expanding the path integral action (118) around the classical path requires the expansion of its associate Lagrangean (146); so we get accordingly

With the action (147b) one observes it practically

○ The classical action goes outside of the path integration by simply becoming the multiplication factor exp[(_{cl}

○ Since the remaining contribution since depends only on quantum fluctuation

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 (_{b}_{a}

○ 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

○ The resulting working path integral of the propagator now simply reads

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 [

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 [

As such, combining the stipulated propagator properties, one starts by equivalently writing

Now, the propagators from (153) may be written immediately under the general form (152)

Next, assuming the notation

With expression (160) the propagator (152) is fully expressed in terms of classical action as

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 ∂_{cl}_{b}_{b}_{a}_{a}_{a}_{b}

○ In the case the formula (161b) is applicable,

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.

Given a free particle with the Lagrangean

Replacing solution (165b) back in Lagrangean (163) the classical action is immediately found

Next, the quantity (158) is firstly evaluated in the spirit of (161b) as
_{b}

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

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 [

Yet, these cases appeal the spectral representation of the quantum propagators and will not be treated here, being more suited for a dedicated monograph [

The characteristic Lagrangean of the harmonic oscillator

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

On the other hand the classical action of the Lagrangean (172) looks like

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 _{b}_{a}

Note that the correctness of

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
_{b}

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

Thus we have to adjust the propagator (183) with the exponential pre-factor corrected with the complex factor “

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)

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

Remarkably, the result (192) recovers also the energy quantification of the quantum motion under the harmonic oscillator influence, as seen by the successive transformations

When comparing the expression (193a) with the canonical formulation of the partition function

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

○ In computation of the path integral propagator the workable

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 derivation of the evolution amplitude employs some of the previously Feynman path integral ideas refined due to the works of Kleinert and collaborators [

○ The real time dependency is “rotated” into the imaginary time

○ The quantum paths of (145a) are re-parameterized as

In these conditions the quantum statistical path integral representation of quantum propagator becomes

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

Now, looking on _{SC}

Therefore, the semiclassical form of path integral representation of evolution amplitude looks like

The remaining problem is that of expressing the averaged values of the fluctuation paths in single or multiple time connection, _{i}_{i}_{j}_{i}_{j}

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

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 _{+} being the Euclidian action, see

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 (

_{1})⋯_{n}

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

Now, going to the second order of correlation of (213) one has

In similar manner, while applying a kind of recursive rule, sometimes denoted as the

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

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
_{a}_{b}

With these, the harmonic fluctuation action of (228) may be reconsidered with the working form

Finally, back with the identifications

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.

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

Therefore, with the quantities of

Solving the associate real time harmonic problem;

Rotating the solution into imaginary time picture;

Taking the “free harmonic limit”

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

The result (175a) is combined with (177a) to provide the real time classical path

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

The “free-harmonic” (

This gives

The result (237) is implemented in the formula (197) to finally produce the classical fluctuation path

Now, going to the evaluation of the expression (233b) we need the Green function of the harmonic oscillator from the

In the same manner the temporal alternative ordering problem of (240)

Now, looking for appropriate identification in the inhomogeneous equation

It leaves with the real time Green function solution of the harmonic oscillator

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

Expression (248) is finally employed to the “free harmonic” limit (236) providing the result
_{ω→}_{0} (

Returning to evaluate the second order truncated expansion (208) one needs the evaluation of the quantities 〈_{i}_{i}_{j}_{i}_{j}

With the help of expression (238) the first order averaged fluctuation integral appearing on (208) becomes

Going now to the double connected correlation functions, one has the working analytical expression

Now, the second order averaged fluctuation integrals are computed as following:

At coincident times

At different times

Finally, while replacing the values of

Note that the expression (257) plays the role of the semiclassical canonical density in PI-DFT algorithm given by

At this point, the expression (260) may be elegantly transformed through considering the Gauss theorem of integrated divergence that written in a general

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

Here, we assess electronegativity (EN) as the convolution of the imaginary time conditional probability (

Having an analytical EN quantum formulation, the chemical hardness,

The general radial one-dimensional probability amplitude connecting the space-time events (_{a}_{b}_{b}_{a}_{eff}

▪ 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 (_{a}_{b}

Therefore, the electronegativity can be seen also as

Next, within the Bohr description of the electrons moving in a central potential [^{2} / 4_{0} = 1, further atomic dependency is acquired by the Bohr-Slater quantifications
_{eff}_{a}

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 [

According with the exposed strategy, one finds the fourth order semiclassical expansions for electronegativity is given [

Aiming to unfold the electronegativity and chemical hardness atomic scales, through applying the

The numerical fourth order semiclassical electronegativity and chemical hardness atomic scales are reported in

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

However, this rule, while being not always obeyed for the finite difference ^{FD}^{FD}

Going to electronegativity discussion, the present

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

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

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

Finally, we have to point out that the systematic decrease of orders of magnitude of electronegativity and hardness semiclassical scales of

As previously shown, see

Therefore, the associated periodic propagator (density matrix) becomes

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

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

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.

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,

Going to characterize the periodic paths, they will be seen as the Fourier series [_{m}

Moreover, under the condition the quantum paths (282a) are real

With this, the quantified form of periodic path frequencies, ^{th} terms viewed more than the “zero-oscillating” or free motion path but the thermal averaged path over entire quantum paths (282a)

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.

The quantum path decomposition (282d) imposes the factorization of the Feynman path integral measure (120) accordingly
_{0} and _{m}

To this end, let’s begin with the computation of the kinetic term appearing under the integral (288a)

Next, with the frequency choice

With:

Going back, once calculated, the Riemann series (296) is replaced in (295) which, at its turn, is employed for the integral evaluation

Computation of the generalized Riemann series (296) requires few intermediate operations:

○ Writing it under the form

○ Applying the Poisson formula, see

○ Computing the integral under the sum of (299) by the complex integration, according with the contours of integration identified in

○ 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

○ With expression (303) the Riemann series (298a) finally reads as

○ 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

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.

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):

○ Evaluating the function (294) by the aid of (297) rule through considering the variable change

○ Obtaining the function (293a) with the help of (293b) and (310)

○ Releasing the Matsubara partition function for the harmonic motion by replacing function (311) into expression (292)

○ Comparing the form (312) with the consecrated results (192) or (288b), thus getting the condition

○ Choosing for the Feynman centroid normalization factor the inverse of the thermal length (280)

○ Plugging expression (314) in (313) to yield the constant

○ 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.

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

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 _{FK}

In

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 (_{0})^{2}, driven harmonically by the frequency Ω^{2} (_{0}), with a role in optimizing the quantum fluctuations in order state equilibrium be achieved. The supplementary Feynman-Kleinert perturbation function _{FK}_{0}) 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

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

The optimization of the Feynman-Kleinert partition function (322) is performed employing the Jensen-Peierls inequality

When rewritten the last inequality with the help of Euclidian actions for general partition function and the Feynman-Kleinert specialization,

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

Expression (329) can be even more simplified when solving out the

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 ^{2}(_{0}) that accounts for the existing quantum fluctuation in the system.

Nevertheless,

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

When replaced in average form (332) the forms (333) and (334) cumulate with the smeared out potential (331) in the final Feynman-Kleinert average

With

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
^{2})Ξ = (1/ Ω)(∂/ ∂Ω)Ξ the first term in (338) is arranged to emphasize on its vanishing nature when recalling the

This way, from

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.

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.

In order to check (342) one separately computes each of its sides by the aid of

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:

It allows the additional similar relationships

Note that the equality (344b) is due to the symmetry of the smearing average formula (331) at the interchange _{0}, while the mixed term of (344a) expansion vanishes, 〈_{0}〈 _{a2(x0)} = 0, in any path representation. With these rules, one can reconsider

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

_{0}, which goes to the celebrated Hydrogen Coulomb central potential in the limit