Path Integrals for Electronic Densities, Reactivity Indices, and Localization Functions in Quantum Systems
Abstract
:1. Introduction
2. From Density Matrix to Path Integral
2.1. On Mono-, Many-, and Reduced- Electronic Density Matrices
- ○ Normalization
- ○ Recursion
- ○ First order Löwdin reduction
- ○ The idempotency
- ○ The normal additivity, see Equations (33)
- ○ Kernel multiplicity
- ○ Many-body normalization
2.2. Canonical Density, Bloch Equation, and the Need of Path Integral
- ○ It identifies the evolution operator
- ○ It produces the so called Bloch equation [21] by taking its β derivative
- ○ Fulfills the (short times, higher temperature) so called Markovian limiting condition
3. Feynman’s Path Integral of Evolution Amplitude
3.1. Construction of the General Path Integral
- ○ 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
3.2. Schrödinger Equation from Path Integral
3.2.1. Propagator’s Equation
3.2.2. Wave Function’s Equation
3.3. Calculation of Path Integrals. Basic Applications
3.3.1. Path Integrals’ Properties
- 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
- 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).
- ○ 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
- ○ 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 factor
- ○ 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
- ○ The resulting working path integral of the propagator now simply reads
- ○ 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
3.3.2. Path Integral for Free Particle
3.3.3. Path Integral for Harmonic Oscillator
- ○ 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
4. Semiclassical Path Integral of Evolution Amplitude
4.1. Semiclassical Expansion
- ○ The real time dependency is “rotated” into the imaginary time
- ○ The quantum paths of (145a) are re-parameterized as
4.2. Connected Correlation Functions
4.3. Classical Fluctuation Path and Connected Green Function
- Solving the associate real time harmonic problem;
- Rotating the solution into imaginary time picture;
- Taking the “free harmonic limit”ω → 0.
4.3.1. Calculation of Classical Fluctuation Path
4.3.2. Calculation of the Connected Green Function
4.4. Second Order Semiclassical Propagator, Partition Function and Density
- At coincident times
- At different times
4.5. Fourth Order Semiclassical Electronegativity and Chemical Hardness
- ▪ 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).
- 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;
5. Effective Classical Path Integral of Evolution Amplitude
5.1. Effective Classical Partition Function
5.2. Periodic Path Integrals
5.2.1. Matsubara Frequencies and the Quantum Periodic Paths
5.2.2. Matsubara Harmonic Partition Function
5.2.3. The Generalized Riemann’ Series
- ○ Writing it under the form
- ○ Applying the Poisson formula, see Appendix (A8), on series (298b)
- ○ 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
- ○ 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
5.2.4. Periodic Path Integral Measure
- ○ 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 z = ħβΩ/ 2 in (309)
- ○ 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)
5.3. Feynman-Kleinert Variational Formalism
5.3.1. Feynman-Kleinert Partition Function
5.3.2. Feynman-Kleinert Optimum Potential
5.3.3. Quantum Smeared Effects and the Stability of Matter
5.3.4. Ground State (β→∞, T→0K) Case
5.3.5. Excited State (β→0, T→∞) Case. Wigner Expansion
5.4. Path Integral Connection with Density Functional Theory
5.4.1. Feynman-Kleinert Electronic Density. Analogy with Levy’s Search Mechanism
- ○ The variational approach for effective-classical potential partition function provides the energy approximation for the ground state, leaving with the Feynman-Kleinert potential, see Equations (328) and (337);
- ○ The optimization of the Feynman-Kleinert potential respecting the trial harmonic frequency to achieving the thermodynamical equilibrium of the ground state, see Equations (338) & (341).
5.4.2. Mulliken Density Functional Electronegativity
5.4.3. Atomic Electronegativity by Feynman Centroid Path Integral
- ○ The first one considers only the pseudo-potentials into the path integral formalism that gives the electronic density in the quantum statistical manner as it was described in the previous Section 5.4.1. This way, a strong physical meaning is assured because all the information about the electronic density and electronegativity are comprised (and dictated) only by the pseudopotential. Yet, the problem that arises in this approach is that the electronic density depends on the β parameter. This parameter will be fixed so that the electronic density to fulfill the path integral normalization condition. Additionally, the search of the β parameter must be done in the semiclassical (high temperature) limit (β → 0) for which the path integral formalism corresponds to the excited (valence) states of atoms.
- ○ The second approach takes beyond to the pseudopotential data also the valence basis and the electronic densities are then computed in the accustomed quantum manner. At this point we need to consider the working orbital type for the atomic systems and we will chose the s-basis set because its spherical symmetry.