Operators one-electron

The dipole moment operator is a sum of one-electron operators r , and as such the electronic conlribution to the dipole moment can be written as a sum of one-electron contributions. The eleclronic contribution can also be written in terms of the density matrix, P, as follows ... 

When the states P1 and P2 are described as linear combinations of CSFs as introduced earlier ( Fi = Zk CiKK), these matrix elements can be expressed in terms of CSF-based matrix elements < K I eri IOl >. The fact that the electric dipole operator is a one-electron operator, in combination with the SC rules, guarantees that only states for which the dominant determinants differ by at most a single spin-orbital (i.e., those which are "singly excited") can be connected via electric dipole transitions through first order (i.e., in a one-photon transition to which the < Fi Ii eri F2 > matrix elements pertain). It is for this reason that light with energy adequate to ionize or excite deep core electrons in atoms or molecules usually causes such ionization or excitation rather than double ionization or excitation of valence-level electrons the latter are two-electron events. 

Such operators which collect together all the variable terms involving a particular electron are called one-electron operators. The l/ri2 term is a typical two-electron operator, which we often write... 

It is sometimes useful to recast the equation as the expectation value of a sum of one-electron and pseudo one-electron operators... 

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. 

Boys and Cook refer to these properties as primary properties because their electronic contributions can be obtained directly from the electronic wavefunction As a matter of interest, they also classified the electronic energy as a primary property. It can t be calculated as the expectation value of a sum of true one-electron operators, but the Hartree-Fock operator is sometimes written as a sum of pseudo one-electron operators, which include the average effects of the other electrons. 

The one electron operator h, describes the motion of electron i in the field of all the nuclei, and gy is a two electron operator giving the electron-electron repulsion. We note that the zero point of the energy corresponds to the particles being at rest (Tc = 0) and infinitely removed from each other (Vne = Vee = V n = 0). 

For the one-electron operator only the identity operator can give a non-zero contribution. For coordinate 1 this yields... 

The S matrix contains the overlap elements between basis functions, and the F matrix contains the Fock matrix elements. Each element contains two parts from the Fock operator (eq. (3.36)), integrals involving the one-electron operators, and a sum over... 

To construct the Fock matrix, eq. (3.51), integrals over all pairs of basis functions and the one-electron operator h are needed. For M basis functions there are of the order of of such one-electron integrals. These one-integrals are also known as core integrals, they describe the interaction of an electron with the whole frame of bare nuclei. The second part of the Fock matrix involves integrals over four basis functions and the g two-electron operator. There are of the order of of these two-electron integrals. In conventional HF methods the two-electron integrals are calculated and saved before the... 

Requiring the variation of L to vanish provides a set of equations involving an effective one-electron operator (hKs), similar to the Fock operator in wave mechanics... 

The derivative of the core operator h is a one-electron operator similar to the nucleus-electron attraction required for the energy itself (eq. (3.55)). The two-electron part yields zero, and the V n term is independent of the electronic wave function. The remaining terms in eqs. (10.89), (10.90) and (10.95) all involve derivatives of the basis functions. When these are Gaussian functions (as is usually the case) the derivative can be written in terms of two other Gaussian functions, having one lower and one higher angular momentum. 

Matrix element of a one-electron operator in semi-empirical theory... 

Matrix element of a semi-empirical one-electron operator, usually... 

If the Hamiltonian would be the sum of one-electron operators only, one could easily separate the variables in the basic Schrodin-ger equation (Eq. II. 1), and the total wave function 0 would then be the product of N one-particle functions each one being an... 

In a nutshell, the d electrons repel the bonding electrons. They get in the way of the bonds and, to a greater or lesser degree, frustrate the attraction between metal and ligands. In essence, the proposed minimal overlap of d orbitals with the ligands, but significant repulsive interaction with the bonds, is equivalent to a focus upon the two-electron operator rather than the one-electron operator that is, upon repulsions rather than overlap. 

Here, an effective one-electron operator matrix has Fock and energy-dependent, self-energy terms. Prom this matrix expression, one may abstract one-electron equations in terms of the generalized Fock and energy-dependent, self-energy operators ... 

The most important contributions to the spin Hamiltonian can be expressed as one-electron operators, and it will be shown that tl matrices Hf and Hf, vanish, as long as the reference state is computed up to one order of perturbation smaller than these matrices. Thus,... 

In the random phase approximation, the transition amplitude from state 0) to l) for any one electron operator O may be written as... 

The terms involving the commutator arise as one-electron operators do not commute with the exchange operator. Using the properties of JT and /C, from eqs. 7 and 8 it can be shown that... 

Equation 17 can be viewed as the general form of a sum rule for an arbitrary one-electron operator O expressed in terms of the square of the transition moment of the operator and its excitation energies. 

Furthermore, if (t>k are eigenvectors of some one-electron operator h(v) such that ... 

The first term is the familiar one-electron operator, the second term represents the Coulomb potential, and the third term is called exchange-correlation potential. HF and DFT differ only in this last term. In HF theory there is only a nonlocal exchange term, while in DFT the term is local and supposed to cover both exchange and correlation. It arises as a functional derivative with respect to the density ... 

In a rigorous treatment, one replaces the one-electron operator h by the four-component Dirac-operator hjj and perhaps supplement the two-electron operator by the Breit interaction term [15]. Great progress has been made in such four-component ab initio and DPT methods over the past decade. However, they are not yet used (or are not yet usable) in a routine way for larger molecules. 

These N equations have the appearance of eigenvalue equations, where the Lagrangian multipliers are the eigenvalues of the operator f. The have the physical interpretation of orbital energies. The Fock operator f is an effective one-electron operator defined as... 

The first two terms are the kinetic energy and the potential energy due to the electron-nucleus attraction. V HF(i) is the Hartree-Fock potential. It is the average repulsive potential experienced by the i th electron due to the remaining N-l electrons. Thus, the complicated two-electron repulsion operator l/r in the Hamiltonian is replaced by the simple one-electron operator VHF(i) where the electron-electron repulsion is taken into account only in an average way. Explicitly, VHF has the following two components ... 

Since the Fock operator is a effective one-electron operator, equation (1-29) describes a system of N electrons which do not interact among themselves but experience an effective potential VHF. In other words, the Slater determinant is the exact wave function of N noninteracting particles moving in the field of the effective potential VHF.5 It will not take long before we will meet again the idea of non-interacting systems in the discussion of the Kohn-Sham approach to density functional theory. 

The term in square brackets defines the Kohn-Sham one-electron operator and equation (7-1) can be written more compactly as... 

H is the one-electron operator and i is the Slater basis set function (2s, 2p). The diagonal elements of Htj (Hit) are approximated as the valence state ionization potentials and the off-diagonal elements Htj are estimated using the Wolfsberg-Helmholtz approximation,... 

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