Order density

Simons J 1971 Direct calculation of first- and second-order density matrices. The higher RPA method J. Chem. Phys. 55 1218-30... 

The resulting similarity measures are overlap-like Sa b = J Pxi ) / B(r) Coulomblike, etc. The Carbo similarity coefficient is obtained after geometric-mean normalization Sa,b/ /Sa,a Sb,b (cosine), while the Hodgkin-Richards similarity coefficient uses arithmetic-mean normalization Sa,b/0-5 (Saa+ b b) (Dice). The Cioslowski [18] similarity measure NOEL - Number of Overlapping Electrons (Eq. (10)) - uses reduced first-order density matrices (one-matrices) rather than density functions to characterize A and B. No normalization is necessary, since NOEL has a direct interpretation, at the Hartree-Fodt level of theory. 

Comparatively little space will therefore be devoted to some rather recent approaches, such as the plasma model of Bohm and Pines, the two-body interaction method developed by Brueckner in connection with nuclear theory, Daudel s loge theory, and the method of variation of the second-order density matrix. This does not mean that these methods would be less powerful or less impor-... 

In order to evaluate the expectation value of the energy for an electronic system it is hence sufficient to know the generalized second-order density matrix r(x x 2 x1x2), from which the first-order density matrix may be obtained by using the formula... 

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. 

In conclusion it should be added that, during the last few years, even methods for the direct evaluation of the first-order density matrix p(xlt x2) have been developed (McWeeny, 1956). 

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

In Section II.C we gave a general discussion of the Coulomb correlation, and we will now define the correlation error in the independent-particle model in greater detail. It is convenient to study the first- and second-order density matrices and, according to the definitions (Eq. II.9) applied to the symmetryless case, we obtain... 

A certain answer may be found (Lowdin 1955) by considering the first-order density matrix y(x x1) defined by Eq. II.9. By means of the basis y>k and formula III. 14, this matrix may be expressed in the form... 

This theorem follows from the antisymmetry requirement (Eq. II.2) and is thus an expression for Pauli s exclusion principle. In the naive formulation of this principle, each spin orbital could be either empty or fully occupied by one electron which then would exclude any other electron from entering the same orbital. This simple model has been mathematically formulated in the Hartree-Fock scheme based on Eq. 11.38, where the form of the first-order density matrix p(x v xx) indicates that each one of the Hartree-Fock functions rplt y)2,. . ., pN is fully occupied by one electron. 

In Section II.B, we have used the density matrices to simplify the calculations, but the wave functions W are still the fundamental quantities. Relation II. 11 shows,however, that the expectation value of the energy p)Av depends only on the second-order density matrix, and we can rewrite it in the form22... 

Next, we calculate the derived first-order densities (sometimes called marginal densities)... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

Density functions can be obtained up to any order from the manipulation of the Slater determinant functions alone as defined in section 5.1 or from any of the linear combinations defined in section 5.2. Density functions of any order can be constructed by means of Lowdin or McWeeny descriptions [17], being the diagonal elements of the so called m-th order density matrix, as was named by Lowdin the whole set of possible density functions. For a system of n electrons the n-th order density function is constructed from the square modulus of any n-electron wavefunction attached to the n-electron system somehow. 

Using a unnormalized n-electron Slater determinant D(j) as system wavefunction, constructed as discussed in section 5.1, then one can write the n-th order density function p " (j) as ... 

Generalization of this one determinant function to linear combinations of Slater determinants, defined for example as these discussed in the previous section 5.2, is also straightforward. The interesting final result concerning m-th order density functions, constructed using Slater determinants as basis sets, appears when obtaining the general structure, which can be attached to these functions, once spinorbitals are described by means of the LCAO approach. 

The equation (28) has the same structure as the well known LCAO form of the first order density function [9], Thus, it can be concluded that density functions of any order exhibit the same formal structure. In this manner, it can be seen that NSS s lead to an interesting mnemotechnical rule. 

For instance, the first-order density matrix can be written... 

Accordingly, only the projections over the subspace of virtual orbitals are needed to compute the second-order density matrix. 

Eventually one finds the final transformation law for the second-order density matrices... 

Levy, M., 1979, Universal Variational Functionals of Electron Densities, First Order Density Matrices, and Natural Spin Orbitals and Solution of the v-Representability Problem , Proc. Natl. Acad. Sci. USA, 16, 6062. 

Gritsenko, O. V., R. van Leeuven, and E. J. Baerends. 1996. Molecular exchange-correlation Kohn-Sham potential and energy density from ab initio first- and second-order density matrices Examples for XH (X = Li, B, F). J. Chem. Phys. 104, 8535. 

The usual reactivity indices, such as elements of the first-order density matrix, are also incapable of distinguishing properly between singlet and triplet behavior. Recently, French authors 139,140) have discussed the problem and shown how electron repulsion terms can be introduced to obtain meaningful results. The particular case of interest to them was excited state basicity, but their arguments have general applicability. In particular, the PMO approach, which loses much of its potential appeal because of its inability to distinguish between singlet and triplet behavior 25,121) coui(j profit considerably from an extension in this direction. 119,122)... 

In contrast, the NBO and NRT methods make no use of molecular geometry information (experimental or theoretical), but instead provide optimal descriptions of orbital composition or electron-density distributions based directly on the first-order density operator. For this reason the NBO/NRT indices have predictive utility for a broad range of chemical phenomena, without bias toward geometry or other particular empirical properties. 

An order density is a demand density 5 with 5(0) = 0. The number of orders per interval can be described by a discrete density function t] with discrete probabilities defined for nonnegative integers 0,1, 2, 3, —The resulting (t], 8)-compounddensity Junction is constructed as follows A random number of random orders constitute the random demand. The random number of orders is r -distributed. The random orders are independent from the number of orders, and are independent and identically 5-distributed. 

An alternative approach was offered by Lee, Yang, and Parr [19], who derived a gradient-corrected correlation functional ( LYP ) from the second-order density matrix in HF theory. Together with PW91, this functional is currently the most widely used correlation functional for molecular calculations. 

Beginning way back in the 20s, Thomas and Fermi had put forward a theory using just the diagonal element of the first-order density matrix, the electron density itself. This so-called statistical theory totally failed for chemistry because it could not account for the existence of molecules. Nevertheless, in 1968, after years of doing wonders with various free-electron-like descriptions of molecular electron distributions, the physicist John Platt wrote [2] We must find an equation for, or a way of computing directly, total electron density. [This was very soon after Hohenberg and Kohn, but Platt certainly was not aware of HK by that time he had left physics.]... 

Of course, the Coulomb interaction appears in the Hamiltonian operator, H, and is often invoked for interpreting the chemical bond. However, the wave function, l7, must be antisymmetric, i.e., must satisfy the Pauli exclusion principle, and it is the only fact which explains the Lewis model of an electron pair. It is known that all the information is contained in the square of the wave function, 1I7 2, but it is in general much complicated to be analyzed as such because it depends on too many variables. However, there have been some attempts [3]. Lennard-Jones [4] proposed to look at a quantity which should keep the chemical significance and nevertheless reduce the dimensionality. This simpler quantity is the reduced second-order density matrix... 

Now, using Equation 20.4, one can construct the reduced second-order density matrix. For / cr2, P2>det is quite boring ... 

The calculation of the indices requires the overlap matrix S of atomic orbitals and the first-order density (or population) matrix P (in open-shell systems in addition the spin density matrix Ps). The summations refer to all atomic orbitals /jl centered on atom A, etc. These matrices are all computed during the Hartree-Fock iteration that determines the molecular orbitals. As a result, the three indices can be obtained... 

Here, Flffl are matrix elements of a zeroth-order Hamiltonian, which is chosen as a one-electron operator in the spirit of MP2. is an overlap matrix The excited CFs are not in general orthogonal to each other. Finally, Vf)(i represents the interaction between the excited function and the CAS reference function. The difference between Eq. [2] and ordinary MP2 is the more complicated structure of the matrix elements of the zeroth-order Hamiltonian in MP2 it is a simple sum of orbital energies. Here H is a complex expression involving matrix elements of a generalized Fock operator F combined with up to fourth-order density matrices of the CAS wave function. Additional details are given in the original papers by Andersson and coworkers.17 18 We here mention only the basic principles. The zeroth-order Hamiltonian is written as a sum of projections of F onto the reference function 0)... 

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