Operator electron-nuclear attraction

In the latter expression, V(r,) is the electron-nuclear attraction operator describing the interaction of the z th electron of the molecule with the set of nuclei. 

V is the Laplacian operator for the th electron vS( is the th electron-nuclear attraction term... 

Ri,R2,. ..,Rk denotes the nuclear coordinates. The first two terms in equation (1) describe, respectively, the electronic kinetic energy and electron-nuclear attraction and the third term is a two-electron operator that represents the electron-electron repulsion. These three operators comprise the electronic Hamiltonian in free space. The term V(r) is a generic operator for an external potential. One of the common ways to express V(f), when it is affecting electrons only, is to expand it as a sum of one-electron contributions... 

Here h are the one-electron integrals including the electron kinetic energy and the electron-nuclear attraction terms, and gjjkl are the two-electron repulsion integrals defmed by (3 19). The summations in (3 24) are over the molecular orbital basis, and the definition is, of course, only valid as long as we work in this basis. Notice that the number of electrons does not appear in the defmition of the Hamiltonian. All such information is found in the Slater determinant basis. This is true for all operators in the second quantization formalism. 

In the case of k = 0 (like the kinetic energy + electron-nuclear attraction operator in atoms, H(1)), the 6j-symbol collapses to a number... 

Next we can dispose of the matrix elements of the one-electron operator Hy,Eq. (3-3 b) the kinetic energy operator, the electron-nuclear attraction potential arising from the metal nucleus, and the spin-independent relativistic terms have spherical symmetry and can be treated through the definition of the basis orbitals ip, Eq. (3-11). The spin-... 

Here P denotes the density matrix, h contains kinetic energy and electron-nuclear attraction operators [pqllr] and [rllt] are Coulomb repulsion integrals with 3 and 2 indices, respectively, [pqs] denote one-electron 3-index integrals and is the nuclear-nuclear repulsion term. The form of Equation 4 ensures (11b) that the Coulomb energies are accurate up to second order in the difference between the fitted density and the "exact" density obtained directly from the wavefunctioa... 

Here, hi is the operator for kinetic and electron nuclear attraction energies, and / 2 is the two electron operator (l/r,). Since the Gaussian orbitals in eqn (1-A-l) form a non-orthogonal basis set, the following transformation can be used in order to convert it to an orthogonal basis set ... 

The ligand-field operator is a Hamiltonian containing the kinetic energy of the electrons plus three types of potential energy terms, the nuclear-nuclear and electron-electron repulsion terms and the electron-nuclear attraction terms. It may be defined as the Hamiltonian of the central-ion plus ligand system minus that of the central-ion system itself. It may be written as Hi — H2, ... 

VeN (r. R) is the Coulomb electron-nuclear attraction energy operator,... 

Since all the wavefunctions in family 4>p yield the same electron density p(r), they must also have the same expectation value for any local multiplicative operator for example, for the V(i) V ri) electron-nuclear attraction operator, as... 

Of course, this wavefunction Fp.min also gives the same p(r)-determined value of the expectation value for the electron-nuclear attraction operator as that given by Eq. 17 ... 

The appearance of wave-function forces in the expression of 8E is a big change. Consider, for instance, the variation of the energy as a function of a nuclear coordinate. Using the first quantized Hamiltonian and the Hellmann-Feynman theorem, it is clear that only the electron-nuclear attraction operator has a contribution ... 

The operator hi is a one-electron operator, representing the kinetic energy of an electron and the nuclear attraction. The operators J and K are called the Coulomb and exchange operators. They can be defined through their expectation values as follows. 

Consider now the Hamilton operator. The nuclear-nuclear repulsion does not depend on electron coordinates and is a constant for a given nuclear geometry. The nuclear-electron attraction is a sum of terms, each depending only on one electron coordinate. The same holds for the electron kinetic energy. The electron-electron repulsion, however, depends on two electron coordinates. 

In the previous section we examined the variational result of the two-term wave function consisting of the covalent and ionic functions. This produces a 2 x 2 Hamiltonian, which may be decomposed into kinetic energy, nuclear attraction, and electron repulsion terms. Each of these operators produces a 2 x 2 matrix. Along with the overlap matrix these are... 

Let us examine the Schrodinger equation in the context of a one-electron Hamiltonian a little more carefully. When the only terms in the Hamiltonian are the one-electron kinetic energy and nuclear attraction terms, the operator is separable and may be expressed as... 

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