Approximate Density Matrices

Although it is not directly related to the Hellmann-Feynman theorem, we list also the second derivatives of the energy for an approximate wavefunction here 

For a variational energy the last term vanishes again and the second derivative of the energy for variational methods is given as 

In Section 2.3 the electron density P( and a reduced one-electron density matrix P r,r ) were defined in Eqs. (2.17), (2.20) and (2.22). In Section 3.5 it was then shown how the electron density can be used in the calculation of expectation values. 

In the present section we want to derive now approximations to the electron density and reduced one-electron density matrix using two of the approximate wavefunctions presented in the previous sections the SCF wavefunction and the Mpller- 

Plesset perturbation theory wavefunction through second order, 

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Matrix Assembler (ADMA) method, introduced for the generation of ab initio quality approximate density matrices for macromolecules [142-146], and for the computation of approximate macromolecular forces [146], among other molecular properties. 

Mezey PG. Quantum similarity measures and Lowdin s transform for approximate density matrices and macromolecular forces. Int J Quant Chem 1997 63 39-48. 

The approximate density matrix p R,R ) satisfies the translation-symmetry conditions. This matrix also satisfies the point-symmetry condition, provided that the whole star of each vector kj is included in the set of special points fey. However, the ap>-proximate density matrix p R,R ) does not satisfy other properties of the exact DM. It is easy to see that an arbitrary finite sum over vectors kj, of delocaUzed Bloch functions does not decrease with r —> oo. Therefore, provided that the R... 

The incorrect asymptotic behavior of the approximate density matrix p gives rise to divergences in calculations of the average values of some physical quantities. In particular, for the exchange energy per unit ceU Kex, we obtain the divergence... 

Interpolation Procedure for Constructing an Approximate Density Matrix for Periodic Systems... 

According to (4.154), the approximate density matrix p Rn) found by interpolation in the BZ contains the weighting frmction of (4.155)-(4.157) as a factor. This function ensures the proper behavior of the offdiagonal elements of the approximate DM as i2 —> 00. As already mentioned, the matrix without a weighting factor is a periodic (not vanishing at infinity) function... 

For the approximate DM to have the proper point symmetry, the LUC should be taken to be the Wigner-Seitz (WS) cell. In this case, however, the symmetry can be broken if on the boundary of the WS cell, there are atoms of the crystal. Indeed, if an atom lies on the WS cell boundary, then there is one or several equivalent atoms that also lie on the boundary of the cell and their position vectors differ from that of the former atom by a super lattice vector A. When constructing the approximate density matrix p we assigned only one of several equivalent atoms to the WS ceU. In other words, in the set there are no two vectors that differ from each other by a... 

The study of the approximate density matrix properties allowed the implementation of the cyclic cluster model in the Hartree- Fock LCAO calculations of crystalUne systems [100] based on the idempotency relations of the density matrix. The results... 

The sum (6.50) can be calculated for k kj, for example, by the Ewald method. However, for k = kj the series (6.50) appears to be divergent [95]. This divergence is the result of the general asymptotic properties of the approximate density matrix calculated by the summation over the special poits of BZ (see Sect. 4.3.3). The difficulties connected with the divergence of lattice sums in the exchange part have been resolved in CNDO calculations of solids by introduction of an interaction radius... 

In the presence of a potential function U(x,y), the density matrix in the high-temperature approximation has the fomi... 

The CNDO/INDO, MINDO/3, Z3NDO/1, and ZINDO/S methods might be expected to imply an even simpler equation for the electron density than the above. For example, a rigorous complete neglect of CNDO approximation, suggests that equations (87) and (88) should be replaced by expressions with a sum only over diagonal elements of the density matrix. This would represent a molecular charge density that is the exact sum of atomic densities. Alter-... 

Since the exact density matrix is not known, the (approximate) density is written in terms of a set of auxiliary one-electron functions, orbitals, as... 

Defining ethane, ethylene and acetylene to have bond orders of 1, 2 and 3, the constant a-is found to have a value of approximately 0.3 A. For bond orders less than 1 (i.e. breaking and fonning single bonds) it appears that 0.6 A is a more appropriate proportionality constant. A Mulliken style measure of the bond strength between atoms A and B can be defined from the density matrix as (note that this involves the elements of the product of the D and S matrices). 

Population analysis with semi-empirical methods requires a special comment. These methods normally employ the ZDO approximation, i.e. the overlap S is a unit matrix. The population analysis can therefore be performed directly on the density matrix. In some cases, however, a Mulliken population analysis is performed with DS, which requires an explicit calculation of the S matrix. 

These relations show that the Fock-Dirac density matrix is identical with the first-order density matrix, and that consequently the first-order density matrix determines all higher-order density matrices and then also the entire physical situation. This theorem is characteristic for the Hartree-Fock approximation. 

The second-order density matrix is in the Hartree-Fock approximation given by Eqs. 11.44 and 11.53, and we obtain directly... 

Husimi, K., Proc. Phys.-Math. Soc. Japan 22, 264, "Some formal properties of the density matrix." Introduction of the concept of reduced density matrix. Statistical-mechanical treatment of the Hartree-Fock approximation at an arbitrary temperature and an alternative method of obtaining the reduced density matrices are discussed. 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

The treatment presented so far is quite general and formally exact. It combines the eikonal representation for nuclear motions and the time-dependent density matrix in an approach which could be named as the Eik/TDDM approach. The following section reviews how the formalism can be implemented in the eikonal approximation of short wavelengths for the nuclear motions, and for specific choices of electronic states leading to the TDHF equations for the one-electron density matrix, and to extensions of TDHF. 

The correction to the relaxing density matrix can be obtained without coupling it to the differential equations for the Hamiltonian equations, and therefore does not require solving coupled equations for slow and fast functions. This procedure has been successfully applied to several collisional phenomena involving both one and several active electrons, where a single TDHF state was suitable, and was observed to show excellent numerical behavior. A simple and yet useful procedure employs the first order correction F (f) = A (f) and an adaptive step size for the quadrature and propagation. The density matrix is then approximated in each interval by... 

Approximations have been reviewed in the case of short deBroglie wavelengths for the nuclei to derive coupled quantal-semiclassical computational procedures, by choosing different types of many-electron wavefunctions. Time-dependent Hartree-Fock and time-dependent multiconfiguration Hartree-Fock formulations are possible, and lead to the Eik/TDHF and Eik/TDMCHF approximations, respectively. More generally, these can be considered special cases of an Eik/TDDM approach, in terms of a general density matrix for many-electron systems. 

The Eik/TDDM approximation can be computationally implemented with a procedure based on a local interaction picture for the density matrix, and on its propagation in a relax-and-drive perturbation treatment with a relaxing density matrix as the zeroth-order contribution and a correction due to the driving effect of nuclear motions. This allows for an efficient computational procedure for differential equations coupling functions with short and long time scales, and is of general applicability. 

The fundamental tool for the generation of an approximately transferable fuzzy electron density fragment is the additive fragment density matrix, denoted by Pf for an AFDF of serial index k. Within the framework of the usual SCF LCAO ab initio Hartree-Fock-Roothaan-Hall approach, this matrix P can be derived from a complete molecular density matrix P as follows. 

In fact, for a simple, but still remarkably usefi.il first approximation of the electronic density of the new nuclear arrangement K k. one may use the same density matrix Pk ((pKi.)), but in combination with a new basis set cp (K t) obtained by simply moving the centers of the old AO basis functions to the new nuclear locations,... 

This new, approximate macromolecular density matrix (q K ), K [A]) for the new, slightly distorted nuclear geometry K1 is also idempotent with respect to multiplication involving the actual new overlap matrix S(K... 

If the original macromolecular density matrix is already available, then such approximate macromolecular electron densities for slightly distorted nuclear geometries are simpler to calculate than the full recalculation of an ADMA macromolecular density matrix that involves a new fragmentation procedure. 

Having considered only the single mode case so far, we can also derive an expression of x"(copr, t) for a multimode system in a similar fashion. In the twomode case, for instance, %"(apr, x) can be divided into three terms, each of which corresponds to interference between the vibrational processes of the two modes. It should be noted here that within the same approximations as used above, the density matrix of the two modes during the time interval x can be expressed as a product of each mode s matrix. 

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