The first-order density matrix

Having used the generalized Sturmian method to calculate the wave function for an A-electron atom, we are in a position to derive both the corresponding density distribution and the first-order density matrix [36-52], However, because we cannot assume orthonormality between the one-electron spin-orbitals of different configurations, it is necessary to use expressions analogous to the generalized Slater-Condon rules. If we let 

The derivation of equation (42) is similar to that of equation (37). In both cases, one obtains the final result by expanding both and in terms of their minors. Taking the scalar product of equation (42) with both ,(1) and p b ), we obtain 

Here the set of one-electron spin-orbitals q a can be any complete orthonormal set. Summing over configurations, we obtain the first-order density matrix corresponding to the state 

In using equation (44), it is important to remember that in order for the properties of the first-order density matrix to be correct, it is necessary that = 1. This normahzation can be achieved by means of the first generalized Slater-Condon rule, equation (35). 

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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). 

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. 

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. 

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

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)... 

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.]... 

Here we will present the formulae needed for calculating the reduced one-particle density matrices from the floating correlated Gaussians used in this work. The first-order density matrix for wave function T (ri,r2,..., r ) for particle 1 is defined as... 

This long-range correlation effect shows up in both the first-order density matrix and the exchange-correlation hole for finite systems [19]. We concentrate here on the exchange-correlation hole. The general asymptotic form of the pair density is then... 

W. A. Bingel and W. Kutzelnigg, Symmetry of the first-order density matrix and its natural orbitals for linear molecules. Adv. Quantum Chem. 5, 201 (1970). 

The super-CI method now implies solving the corresponding secular problem and using tpq as the exponential parameters for the orbital rotations. Alternatively we can construct the first order density matrix corresponding to the wave function (4 55), diagonalize it, and use the natural orbitals as the new trial orbitals in I0>. Both methods incorporate the effects of lpq> into I0> to second order in tpq. We can therefore expect tpq to decrease in the next iteration. At convergence all t will vanish, which is equivalent to the condition ... 

This expression is actually sometimes used to define the first and second order density matrix. It is anyway useful to know that the first order density matrix elements are equal to the coefficients for the corresponding one-electron integrals in the energy expression and similarly for the second order density matrix elements. The definition of the third order density matrix is,... 

R. Benesch and S. R. Singh, Chem. Phys. Lett., 10, 151 (1971). On the Relationship of the X-Ray Form Factor to the First-Order Density Matrix in Momentum Space. 

Assuming that in a given case the structures of all the participating molecular species is known, it is possible to begin the practical exploitation of Eq. (22) and aim at the variational formulation of the least-motion principle. For this purpose, it is first necessary to introduce the first order density matrix p(9,

position vector of the i-th electron, its spin coordinate and N the total number of electrons... 

The nonlinear nature of the Hamiltonian implies a nonlinear character of the Cl equations which must be solved through an iteration procedure, usually based on the two-step procedure described above. At each step of the iteration, the solvent-induced component of the effective Hamiltonian is computed by exploiting the first-order density matrix (i.e. the expansion Cl coefficients) of the preceding step. In addition, the dependence of the solvent reaction field on the solute wavefunction requires, for a correct application of this scheme, a separate calculation involving an iteration optimized on the specific state (ground or excited) of interest. This procedure has been adopted by several authors [17] (see also the contribution by Mennucci). 

The requirement needed to incorporate the solvent effects into a state-specific (multireference) method is fulfilled by using the effective Hamiltonian defined in Equation (1.159). The only specificity to take into account is that in order to calculate Va we have to know the density matrix of the electronic state of interest (see the contribution by Cammi for more details). Such nonlinear character of Va is generally solved through an iterative procedure [35] at each iteration the solvent-induced component of the effective Hamiltonian is computed by exploiting Equation (1.157) with the apparent charges determined from the standard ASC equation with the first order density matrix of the preceding step. At each iteration n the free energy of each state K is obtained as... 

The natural spin-orbitals, which were first introduced by Lowdin179 in 1955, were shown to produce the most rapidly convergent expansion of the first-order density matrix,180 Thus, the number of determinants needed for any required degree of accuracy in the wavefunction can be greatly reduced if the determinantal wave-functions are constructed from natural orbitals at each stage in the calculation. 

The effectiveness of NSO s in reducing the expansion size in systems with more than two electrons is not as great and, in fact, for larger systems, their use is not practical. The loss in practicality is immediately obvious when one realizes that in order to obtain them, one must diagonalize the first-order density matrix of the exact wavefunction, i.e. a full configuration interaction must first be performed. Two methods have been introduced in order to regain the initial usefulness of natural orbitals the pseudonatural orbital method and the approximate or iterative natural orbital method. 

In the pseudo-natural orbital (PSNO) method,188 a natural orbital calculation is performed on selected pairs of electrons in the Hartree-Fock field of the n-2 electron core. These orbitals are then used as a basis for a Cl calculation. In their work on the HeH+ system, for which the orbital occupancy is 1 first-order density matrix, only 45 configurations were found to have significant occupation numbers. The results of the application of this method to Hs184 and HeH 185 will be discussed below. 

The term containing Dirac s delta ZaS(r — Ra) represents the contribution from the positive point charge Za at the position Ra of the nucleus a, and -pA(r) is the electronic charge distribution, given by the diagonal element of the first-order density matrix normalized to the number of electrons in the monomer A. 

The system electron density p(r) and hence the one-electron probability distribution p(r) = p(r)/N, that is, the density per electron or the shape factor of p, are determined by the first-order density matrix y in the AO representation, also called the charge and bond order (CBO) matrix,... 

In this equation and k denote the standard Coulomb and exchange operators involving core orbital (pi, D and D(ji v) stand for the normalisation integral for the SC wavefunction and the elements of the first-order density matrix in the space of the SC orbitals, and and k, are generalised Coulomb and exchange operators with matrix elements = XikMq )AXp K u Xq) = At least Voutofthe Af... 

The ADMA and SADMA methods generate ab initio quality density matrices P for large molecules M, while avoiding the computation of macromolecular wave functions. At the Hartree-Fock level, the first-order density matrix P fully determines all higher-order density matrices. Within the Hartree-Fock framework, expectation values for one-electron and two-electron operators can be computed using the first-order and second-order density matrices. Consequently, the ADMA and SADMA methods provide new possibilities for adapting quantum-chemical techniques for macromolecules. 

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