Iterative Fock Matrix Construction

To construct the Fock matrix, one must already know the molecular orbitals ( ) since the electron repulsion integrals require them. For this reason, the Fock equation (A.47) must be solved iteratively. One makes an initial guess at the molecular orbitals and uses this guess to construct an approximate Fock matrix. Solution of the Fock equations will produce a set of MOs from which a better Fock matrix can be constructed. After repeating this operation a number of times, if everything goes well, a point will be reached where the MOs obtained from solution of the Fock equations are the same as were obtained from the previous cycle and used to make up the Fock matrix. When this point is reached, one is said to have reached self-consistency or to have reached a self-consistent field (SCF). In practice, solution of the Fock equations proceeds as follows. First transform the basis set / into an orthonormal set 2 by means of a unitary transformation (a rotation in n dimensions),... 

Let s turn our attention to the computational needs of the structure problem (See Fig. 2). Defining n as the number of atomic basis functions employed, there are several characteristic matrices to consider. The Fock matrix, which we repeatedly construct and diagonalize until convergence, is only an n x n matrix unfortunately, to construct this matri at each iteration of the SCF procedure, we have to process the n /8 two-electron integrals. At the current state of the art, the number of atomic basis functions n tends to be about 100. This limitation is not so much because of the difficulty of diagonalizing 100 x 100 matrices, but because of the 10 limitations inherent with processing the tens of millions of integrals at each iteration. 

The crucial problem in the point (ii) - the construction of the Fock matrix - is a retrieval and effective manipulation of the integrals (av >,S ). Since all nonvanishing integrals must be read in from the tape or disk in each iteration, this manipulation may be very time consuming. Programs greatly differ on this point. From the discussion 211,219,227-229 follows that it is profitable to have... 

The generalized Fock matrix F, defined above, is a convenient partial sum because it can also be used in the Hessian matrix element construction. This matrix is not symmetric before convergence is reached in the MCSCF iterative procedure, but becomes symmetric at convergence , resulting in vanishing orbital gradient vector elements. 

As described in Sec. 3.1, each Hartree-Fock iteration involves the construction of the Fock matrix for a given density matrix, followed by the diagonalization of the Fock matrix to generate a set of improved spin orbitals and thus an improved density matrix. Formally, the construction of the Fock matrix requires a number of operations proportional to K4, where K is the number of atoms (because the number of two-electron integrals scales as Al4). For large systems, however, this quartic scaling with K (i.e., with system size) can be reduced to linear by special techniques, as will now be discussed. 

In the SK DFT PBC implementation as mnch work as possible is done in real space. Conseqnently, all matrices are stored in real-space form and transformed into k space only when needed. In the iterative part of the code, first the entire real-space Fock matrix is constructed, it is transformed into several fc-space matrices, which... 

Since the Fock matrix depends on the density matrix, and therefore on the LCAO coefficients themselves, the Roothaan-Hall equations must in general be solved in an iterative fashion. An initial guess of the density matrix is made (typically a diagonal matrix) and a starting Fock matrix F< > is assembled according to equation (6). This Fock matrix is used in equation (8) to arrive at an approximate set of LCAO coefficients which are then inserted into equation (3) to construct a slightly improved density matrix The whole process is repeated until convergence on P is achieved and an SCF is established. 

The unmixed density matrix P is obtained directly from the Roothaan-Hall eigenvectors in iteration k, and the mixed density matrix P is used to construct the latest Fock matrix F via equation (6). A simplified Newton-Raphson technique described by Badziag and Solms is used to determine the optimum value of the scalar parameter otp. This nnixing scheme is applied to the global density matrix in both standard and D C calculations. 

In self-consistent methods, the density matrix, together with the one- and two-electron integrals, is then used in the construction of a new Fock matrix. By iterating, an SCF could be achieved. 

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. 

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