Fock matrix scalar

Bold quantities are operators, vectors, matrices or tensors. Plain symbols are scalars. a Polarizability a, P Spin functions a, p Dirac 4x4 spin matrices ap-jS Summation indices for basis functions F Fock operator or Fock matrix Fy, Eajd Fock matrix element in MO and AO basis Y Second hyperpolarizability yk Density matrix of order k gc Electronic g-factor... 

With these definitions, the component-diagonal blocks of the Fock matrix in the scalar basis are... 

Including the Breit term for the electron-electron interaction in a scalar basis requires extensive additions to a Dirac-Hartree-Fock-Coulomb scheme. It is not possible to achieve the same reductions as for the Coulomb term, and the derivation of the Fock matrix contributions requires considerable bookkeeping. We will not do this in detail, but will provide the development for the Gaunt interaction as we did for the 2-spinor case. 

The decision of whether to work with 2-spinors or a scalar spin-orbital basis must be made at an early stage of computer program construction because it affects all stages of the SCF process evaluation of the integrals, construction of the Fock matrix, and solution of the SCF equations. However, at each stage, the scalar spin-orbital basis can be transformed to the 2-spinor basis. Transformation of the integrals to a 2-spinor basis is not particularly difficult it is similar in principle to the transformation from Cartesians to spherical harmonics. Some efforts have been made to develop new algorithms in which these transformations are incorporated, and RKB is implemented from the start in the 2-spinor basis (Quiney et al. 1999, 2002, Yanai et al. 2002). 

Thus, the scalar basis involves about 20% fewer real quantities than the 2-spinor basis, and therefore less work in the Fock matrix construction. This applies to an uncontracted basis set. 

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 this paper we present the first application of the ZORA (Zeroth Order Regular Approximation of the Dirac Fock equation) formalism in Ab Initio electronic structure calculations. The ZORA method, which has been tested previously in the context of Density Functional Theory, has been implemented in the GAMESS-UK package. As was shown earlier we can split off a scalar part from the two component ZORA Hamiltonian. In the present work only the one component part is considered. We introduce a separate internal basis to represent the extra matrix elements, needed for the ZORA corrections. This leads to different options for the computation of the Coulomb matrix in this internal basis. The performance of this Hamiltonian and the effect of the different Coulomb matrix alternatives is tested in calculations on the radon en xenon atoms and the AuH molecule. In the atomic cases we compare with numerical Dirac Fock and numerical ZORA methods and with non relativistic and full Dirac basis set calculations. It is shown that ZORA recovers the bulk of the relativistic effect and that ZORA and Dirac Fock perform equally well in medium size basis set calculations. For AuH we have calculated the equilibrium bond length with the non relativistic Hartree Fock and ZORA methods and compare with the Dirac Fock result and the experimental value. Again the ZORA and Dirac Fock errors are of the same order of magnitude. 

Any Fock operator can be represented as a sum of the symmetric one and of a perturbation which includes both the dependence of the matrix elements on nuclear shifts from the equilibrium positions and the transition to a less symmetric environment due to the substitution. To pursue this, we first introduce some notations. Let hi be the supervector of the first derivatives of the matrix of the Fock operator with respect to nuclear shifts Sq counted from a symmetrical equilibrium configuration. By a supervector, we understand here a vector whose components numbered by the nuclear Cartesian shifts are themselves 10 x 10 matrices of the first derivatives of the Fock operator, with respect to the latter. Then the scalar product of the vector of all nuclear shifts 6q j and of the supervector hi yields a 10 x 10 matrix of the corrections to the Fockian linear in the nuclear shifts ... 

Here we introduce the notation ( . ..) for the scalar product of vectors whose components are numbered by the Cartesian shifts of the nuclei). Next, let h" be the supermatrix of the second derivatives of the matrix of the Fock operator with respect to the same shifts. As previously, we refer here to the supermatrix indexed by the pairs of nuclear shifts in order to stress that the elements of this matrix are themselves the 10 x 10 matrices of the corresponding second derivatives of the Fock operator with respect to the shifts. The contribution of the second order in the nuclear shifts can be given the form of the (super)matrix average over the vector of the nuclear shifts ... 

The computation times for the evaluation of relativistic Hamiltonians and for one SCF iteration step are presented in Table 14.5. Note that only the nuclear external potential has been considered in the construction of the unitary transformation [cf. Eq. (14.59)]. We see that the computation of the X2C Hamiltonian is slightly faster than that of the BSS Hamiltonian since three additional matrix multiplications are required for the BSS approach, as is evident from Table 14.2. For scalar calculations, the computation time of the X2C Hamiltonian is very close to that of DKH8. The fastest DKH2 approach is about five times faster than the X2C approach (for the setup of the one-electron Hamiltonian). Compared with the computation time of SCF iterations, one Hartree-Fock iteration is about twice as expensive as the X2C transformation. Because several tens of iterations are usually required to obtain converged results, the SCF iterations dominate the total computation time in a Hartree-Fock calculation. The DLU approximation dramatically reduces the computation time. Point-group symmetry can be exploited in scalar-relativistic calculations. As... 

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