Fock particle density matrices

The concept of purification is well known in the linear-scaling literature for one-particle theories like Hartree-Fock and density functional theory, where it denotes the iterative process by which an arbitrary one-particle density matrix is projected onto an idempotent 1-RDM [2,59-61]. An RDM is said to be pure A-representable if it arises from the integration of an Al-particle density matrix T T, where T (the preimage) is an Al-particle wavefiinction [3-5]. Any idempotent 1-RDM is N-representable with a unique Slater-determinant preimage. Within the linear-scaling literature the 1-RDM may be directly computed with unconstrained optimization, where iterative purification imposes the A-representabUity conditions [59-61]. Recently, we have shown that these methods for computing the 1 -RDM directly... 

Here the generalized Fock operator/with matrix elements/ appears for the first time. It looks familiar and resembles the Fock operator of Hartree-Fock theory. However, now the yf are matrix elements of the exact one-particle density matrix... 

In quantum chemistry, the correlation energy Ecorr is defined as Econ = exact HF- In Order to Calculate the correlation energy of our system, we show how to calculate the ground state using the Hartree-Fock approximation. The main idea is to expand the exact wavefunction in the form of a configuration interaction picture. The first term of this expansion corresponds to the Hartree-Fock wavefunction. As a first step we calculate the spin-traced one-particle density matrix [5] (IPDM) y ... 

The static Hartree-Fock problem assumes T-reversal invariance and T-even single-particle density matrix. In this case, Skyrme forces can be limited by only T-even densities psif) Tsif) In the case of dy-... 

A good first approach to a quantum mechanical system is often to consider one-electron functions only, associating one such function, a spin-orbital , with one electron. Most popular are the one-electron functions which minimize the energy in the sense of Hartree-Fock theory. Alternatively one can start from a post-HF wave function and consider the strongly occupied natural spin orbitals (i.e. the eigenfunctions of the one-particle density matrix with occupation numbers close to 1) as the best one-electron functions. Another possibility is to use the Kohn-Sham orbitals, although their physical meaning is not so clear. 

The integral-driven procedure indicated above is practicable only if the elements of the two-particle density matrix can be rapidly accessed. In the closed-shell Hartree-Fock case, the two-particle density matrix can be easily constructed from the one-particle density. The situation is similar for open-shell and small multiconfigurational SCF wavefunctions the two-particle density matrix can be built up from a few compact matrices. In most open-shell Hartree Fock theories (Roothaan, 1960), the energy expression (Eq. (23))... 

The unrestricted Hartree-Fock (UHF) case is completely analogous to the closed-shell one. New terms do appear in the open-shell SCF and the few-configuration case. Nevertheless, the preferred technique is quite similar to the closed-shell case. In particular, the two-particle density matrix can be constructed from compact matrices, and the solution of the derivative Cl equations is very simple, due to the small dimension. 

The operator / has an obvious physical interpretation it is a generalized fock-like operator involving the average potential generated by the one-particle density matrix of tpo- Since we have chosen our orbitals to be natural, the density matrix is diagonal. For the core orbitals, the diagonal elements are unity, while for the valence orbitals they are the occupancy of these orbitals ... 

The Fock operator f and the one-particle density matrix 7 commute, i.e. have common eigenfunctions. This allows an iterative construction of 7 from the eigenstates of f. The leading relativistic corrections for the Dirac-Coulomb operator are ... 

Where the non-correlated energy E c is of the same form as the Hartree-Fock energy, but with 7 the exact one-particle density matrix, which is usually not idempotent. This concept has not yet been applied in practice. 

In the normal (probability theory) use of the term, two probability distributions are not correlated if their joint (combined) probability distribution is just the simple product of the individual probability distributions. In the case of the Hartree-Fock model of electron distributions the probability distribution for pairs of electrons is a product corrected by an exchange term. The two-particle density function cannot be obtained from the one-particle density function the one-particle density matrix is needed which depends on two sets of spatial variables. In a word, the two-particle density matrix is a (2 x 2) determinant of one-particle density matrices for each electron ... 

In fact in this resides the power of the density matrix formalism reducing a many-body problem to the single particle density matrix, abstracted from the single Slater determinant of Eq. (4.190) called also as Fock-Dirac matrix... 

This arbitrariness most clearly manifests itself in going beyond the scope of the HF approximation, as evidenced by a wide variety of definitions for molecular systems available in the literature for valences and bond orders in the case of post-Hartree-Fock methods for molecular systems [570,578-580]. In post-HF methods local characteristics of molecular electronic structure are usually defined in terms of the first-order density matrix and in this sense there is no conceptual difference between HF and post-HF approaches [577]. It is convenient to introduce natural (molecular) spin orbitals (NSOs), i.e. those that diagonalise the one-particle density matrix. The first-order density matrix in the most general case represents some ensemble of one-electron states described by NSOs... 

The Fock space matrix element is called the first-order density matrix or one-particle density matrix. 

Because the Fock matrix depends on the one-particle density matrix P constructed conventionally using the MO coefficient matrix C as the solution of the pseudo-eigenvalue problem (Eq. [7]), the SCF equation needs to be solved iteratively. The same holds for Kohn-Sham density functional theory (KS-DFT) where the exchange part in the Fock matrix (Eq. [9]) is at least partly replaced by a so-called exchange-correlation functional term. For both HF and DFT, Eq. [7] needs to be solved self-consistently, and accordingly, these methods are denoted as SCE methods. 

Now that we have described how to reduce the scaling behavior for the construction of the Coulomb part in the Fock matrix (Eq. [9]), the remaining part within HF theory, which is as well required in hybrid DFT, is the exchange part. The exchange matrix is formed by contracting the two-electron integrals with the one-particle density matrix P, where the density matrix elements couple the two sides of the integral ... 

By diagonalizing the Fock matrix, the canonical MO coefficient matrix (C) is obtained (see Eq. [7]). However, we have seen in a previous section that almost all elements in the coefficient matrix are significant, which contrasts with the favorable behavior of the one-particle density matrix (P). The density matrix is conventionally constructed from the coefficient matrix by a matrix product (Eq. [8]). Although the Roothaan-Hall equations are useful for small-to medium-sized molecules, it makes no sense to solve first for a nonlocal quantity (C) and generate from this the local quantity (P) in order to compute the Fock matrix or the energy of a molecule. Therefore, the goal is to solve directly for the one-particle density matrix as a local quantity and avoid entirely the use of the molecular orbital coefficient matrix. 

As discussed, for a formulation of SCF theories suitable for large molecules, it is necessary to avoid the nonlocal MO coefficient matrix, which is conventionally obtained by diagonalizing the Fock matrix. Instead we employ the one-particle density matrix throughout. For achieving such a reformulation of SCF theory in a density matrix-based way, we can start by looking at SCF theory from a slightly different viewpoint. To solve the SCF problem, we need to minimize the energy functional of... 

For the weakest occupied orbitals, it makes sense to interchange particles and holes. The resulting analog to the MCSCF Fock matrix is then 2f- and the hole density matrix is, as before, d = 2 1 - D. 

Using the block-diagonal nature of the density matrix, besides reducing the requirement in CPU time, also allows one to label the DMEV by the appropriate particle number(ne) and the z— component of the total spin of the block [Mg,a)- The Fock space of the individual sites that are added at each iteration are eigenstates of the site spin and number operators. This allows targeting a definite particle number Nf.) and a definite projected spin (M ) state of the total system. 

---