Core Hamiltonian operator

In the Kohn Sham equations (A.116) [324, 325], the core Hamiltonian operator h( 1) has the same definition as in HF theory (equation A.6), as does the Coulomb operator, 7(1), although the latter is usually expressed as... 

The MNDO v ence-electron Hamiltonian H is given by (16/83) and the Fock matrix elements are given by (16.84). As in (16.85), the core Hamiltonian operator for valence electron 1 is written as H° °(l) = - jV -I- EbVb(1)> where Ya(l) is the part of the potential energy of electron 1 due to its interactions with the core (nucleus plus inner-shell electrons) of atom B. To evaluate the Fock matrix element in (16.84), we... 

This is the usual form of the Hartree-Fock equations. The Fock operator /(I) is the sum of a core Hamiltonian operator h l) and an effective one electron potential operator called the Hartree-Fock potential t fl),... 

In the Fock operator, the core Hamiltonian h( 1) does not depend on the orbitals, but the Coulomb and exchange operators (1) and ( 1) depend on ( 1). If (1,2,3,..., Ne) is constructed from the lowest energy Ne orbitals, one has the lowest possible total electronic energy. By Koopmans theorem, the negative of the orbital energy is equal to one of the ionization potentials of the molecule or atom. 

Hamiltonian operators, 88, 151, 153, 197 core, 204 commutation with O, 200, 218 invarianoe of, 151. harmonic foroe constants, 165. harmonic oscillator, approximation, 165 equation, 170. 

Within the DFT framework, the molecular Kohn Sham (KS) operator for a molecular solute becomes a sum of the core Hamiltonian h, a Coulomb and (scaled) exchange term, the exchange-correlation (XC) potential Vxc and the solvent reaction operator VPCM of Eq. (7-1), namely ... 

Finally, hc i) is the one-electron Hamiltonian operator for electron i in the average field produced by the Nc core electrons,... 

The simplest way to deal with energetically low-lying closed-shell core orbitals is to take them directly from a preceding SCF calculation without further optimization. In this case one has to eliminate all rows and columns corresponding to core orbitals from the matrices P, Q, R, A, B, etc., and replace the one-electron Hamiltonian h by a core Fock operator F. This operator is calculated in the AO basis according to... 

The link to the molecular level of description is provided by statistical thermodynamics whore our focus in Chapter 2 will be on specialized statistical physical ensembles designed spc cifically few capturing features that make confined fluids distinct among other soft condensed matter systems. We develop statistical thermodynamics from a quantum-mechanical femndation, which has at its core the existence of a discrete spectrum of energj eigenstates of the Hamiltonian operator. However, we quickly turn to the classic limit of (quantum) statistical thermodynamics. The classic limit provides an adequate framework for the subsequent discussion because of the region of thermodynamic state space in which most confined fluids exist. 

The theory of symmetry-preserving Kramers pair creation operators is reviewed and formulas for applying these operators to configuration interaction calculations are derived. A new and more general type of symmetry-preserving pair creation operator is proposed and shown to commute with the total spin operator and with all of the symmetry operations which leave the core Hamiltonian of a many-electron system invariant. The theory is extended to cases where orthonormality of orbitals of different configurations cannot be assumed. 

Equations (34) and (35) tell us how many-electron states, constructed by letting electron creation operators act on the vacuum state, transform under the elements of the symmetry group of the core Hamiltonian. The vacuum state is assumed to be invariant under the action of these symmetry operations, i.e., R10) = 10). 

The simple form of this operator is due to the fact that the creation and annihilation operators b bs refer to one-electron spin-orbitals which are eigenfunctions of the core Hamiltonian hc with one-electron energies es. 

In other words, when a Kramers pair creation operator acts on an (N — 2)-electron state 1,4). which is an eigenfunction of the core Hamiltonian, it produces an /V-electron state which is also an eigenfunction of //, with an eigenvalue increased... 

Most of the formalism derived in previous sections for li and By can be carried through with only slight changes for the more general operators W, z and W,, /. For example, the commutation relation with the core Hamiltonian is found to be... 

---