Core-Hamiltonian matrix

In order to form the Fock matrix of an ab initio calculation, all the core-Hamiltonian matrix elements, H y, and two-electron integrals (pvIXa) have to be computed. If the total number of basis functions is m, the total number of the core Hamiltonian matrix elements is... 

The atomic quantities SHn are equal to the perturbations Shaa of the corresponding core Hamiltonian matrix elements in the ligand AO basis. This... 

The atomic quantities ShA are equal to the perturbations Shaa of the corresponding core Hamiltonian matrix elements in the ligand AO basis. This is so because within the CNDO approximation [74] accepted in [58], for the description of the /-system, the quantities 6haa are the same for all aeA. 

In the discussion below, adapted from [22], one can see, for some of the NDO methods, how the core Hamiltonian matrix elements H/(/( (diagonal) and H (off-diagonal)] and electron-electron repulsion integrals /(v are given by several possible combinations of ... 

The orthogonalization transformation ( >- ) for the one-electron integrals causes substantial changes in the core Hamiltonian matrix ... 

From a computational point of view, - H could be evaluated analytically ab initio) and then be added to the semiempirical core Hamiltonian matrix. This procedure, however, introduces an imbalance between the one- and two-electron parts of the Fock matrix as long as the two-electron integrals are not subjected to the same exact transformation (J)), which would sacrifice the computational efficiency of semiempirical methods and is therefore not feasible. Hence the orthogonalization corrections to the one-electron integrals must instead be represented by suitable parametric functions. Their essential features can be recognized from the analytic expressions for the matrix elements of in the simple case of a homonuclear diatomic molecule with two orbitals at atom A, at atom B) ... 

The introduction of orthogonalization corrections for the one-center part of the core Hamiltonian matrix [37] generates improved excitation energies by correcting for deficiencies inherent to the ZDO approximation. 

Under the ZDO approximation the orthogonalization procedure gives rise to significant changes in the core Hamiltonian matrix. There is also an observation that the interactions between neighboring atoms are too large for the usual parametric schemes within the... 

The elements of the core-Hamiltonian matrix are integrals involving the one-electron operator h l describing the kinetic energy and nuclear attraction of an electron, i.e.. 

Calculating the elements of the core-Hamiltonian matrix thus involves the kinetic energy integrals... 

Given a particular basis set the integrals of T and need to be evaluated and the core-Hamiltonian matrix formed. The core-Hamiltonian matrix, unlike the full Fock matrix, needs only to be evaluated once as it remains constant during the iterative calculation. The calculation of kinetic energy and nuclear attraction integrals is described in Appendix A. 

The core-Hamiltonian matrix is the sum of the above three matrices. 

Here F denotes the Fock matrix, F ° the Madelung-corrected Fock matrix, P the density matrix, H is the core Hamiltonian matrix, Etot the total energy of the system and Vmn the nuclear repulsion term. The terms VMAD(P) and denote the... 

Since the ordinary electronic structure calculations yield only ((2ct)> charge-constrained calculations have to be carried out in order to obtain (Qcr) for nonequilibrium values of Qcr- The constraining of charge is easily accomplished at any level of theory by adding terms proportional to the respective sums of AOMs (equation 2) to the core Hamiltonian matrix. " The difference. 

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