One- and two-electron operators

The second-quantization one- and two-electron operators in a nonorthogonal basis may be obtained from the corresponding operators in the auxiliary orthonormal basis discussed in Section 1.4. The 

Inserting this expression in (1.9.17), we obtain the final expression for one-electron operators in a nonorthogonal basis  

We now consider the effect of a one-electron operator / on an ON vector. Combining equations (1.9.15) and (1.9.16), we obtain 

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Using the notation given above for the one- and two-electron operators, the electronic Hamiltonian is... 

The first step is to work out e in terms of the one- and two-electron operators and the orbitals. .., For a polyatomic, polyelectron molecule, the electronic Hamiltonian is a sum of terms representing... 

Expanding out the exponential in eq. (4.46) and using the fact that the Hamilton operator contains only one- and two-electron operators (eq. (3.24)) we get... 

Averages of properties require integrals over CSFs which can readily be written for one- and two-electron operators, insofar the Slater determinants and the MSOs are orthonormal by construction, in terms of one- and two-electron density matrices. 

The first set of equations govern the Cj amplitudes and are called the CI- secular equations. The second set determine the LCAO-MO coefficients of the spin-orbitals (f>j and are called the Fock equations. The Fock operator F is given in terms of the one- and two-electron operators in H itself as well as the so-called one- and two-electron density matrices yij and Tyj i which are defined below. These density matrices reflect the averaged occupancies of the various spin orbitals in the CSFs of VP. The resultant expression for F is ... 

To illustrate how the above developments are carried out and to demonstrate how the results express the desired quantities in terms of the original wavefunction, let us consider, for an MCSCF wavefunction, the response to an external electric field. In this case, the Hamiltonian is given as the conventional one- and two-electron operators H° to which the above one-electron electric dipole perturbation V is added. The MCSCF wavefunction P and energy E are assumed to have been obtained via the MCSCF procedure with H=H°+AV, where A can be thought of as a measure of the strength of the applied electric field. 

To solve Eq. (7.11), we need to know how to evaluate matrix elements of the type defined by Eq. (7.12). To simplify matters, we may note that the Hamiltonian operator is composed only of one- and two-electron operators. Thus, if two CSFs differ in their occupied orbitals by 3 or more orbitals, every possible integral over electronic coordinates hiding in the r.h.s. of Eq. (7.12) will include a simple overlap between at least one pair of different, and hence orthogonal, HF orbitals, and the matrix element will necessarily be zero. For the remaining cases of CSFs differing by two, one, and zero orbitals, the so-called Condon-Slater rules, which can be found in most quantum chemistry textbooks, detail how to evaluate Eq. (7.12) in terms of integrals over the one- and two-electron operators in the Hamiltonian and the HF MOs. 

Since the many-electron Hamiltonian, the effective one-electron Hamiltonian obtained in Hartree-Fock theory (the Fock operator ), and the one- and two-electron operators that comprise the Hamiltonian are all totally symmetric, this selection rule is extremely powerful and useful. It can be generalized by noting that any operator can be written in terms of symmetry-adapted operators ... 

Closely inspecting the operator terms entering the electronic Hamiltonian eq. (1.27) one can easily see that they are sums of equivalent contributions dependent on coordinates of one or two electrons only. Analogously in the second quantization formalism only the products of two and four Fermi operators appear in the Hamiltonian. Inserting the trial. Y-electron wave function of the (ground) state into the expression for the electronic energy yields its expectation value in terms of the expectation values of the one- and two-electron operators ... 

Let us first become familiar with the spin-free (sf) case we have a pair of Slater determinants interacting via (i.e., a sum of spin-independent one-and two-electron operators, h and 12). Their matrix element is given by (Sla-ter-Condon rules) ... 

Since the relativistic many-body Hamiltonian cannot be expressed in closed potential form, which means it is unbound, projection one- and two-electron operators are used to solve this problem [39], The operator projects onto the space spanned by the positive-energy spectrum of the Dirac-Fock-Coulomb (DFC) operator. In this form, the no-pair Hamiltonian [40] is restricted then to contributions from the positive-energy spectrum and puts Coulomb and Breit interactions on the same footing in the SCF calculations. 

Here, hD ) and gf"uIomb are one- and two-electron operators, respectively, a and P are Pauli matrices, c is the speed of light, Nelec is number of electrons, and Vnuc(A) is the nuclear attraction potential. The electron-electron repulsion is assumed to be the Coulomb interaction and electron-positron interactions are disregarded with no pair approximation. 

Using Slater s rules for the expansion of determinants composed of orthonormal orbitals the matrix elements of Eq. (4-6) can be reduced to combinations of orbital matrix elements of one- and two-electron operators. We can write in the usual way... 

The Hamilton operator consists of a sum of one-electron and two-electron operators, eq. (3.24). If two determinants differ by more than two (spatial) MOs there will always be an overlap integral between two different MOs which is zero (same argument as in eq. (3.28)). Cl matrix elements can therefore only be non-zero if the two determinants differ by 0, 1, or 2 MOs, and they may be expressed in terms of integrals of one- and two-electron operators over MOs. These connections are known as the Slater-Condon rules. If the two determinants are identical, the matrix element is simply the energy of a single determinant wave function, as given by eq. (3.32). For matrix elements between... 

The calculation of expectation values of operators over the wavefunction, expanded in terms of these determinants, involves the expansion of each determinant in terms of the N expansion terms followed by the spatial coordinate and spin integrations. This procedure is simplified when the spatial orbitals are chosen to be orthonormal. This results in the set of Slater Condon rules for the evaluation of one- and two-electron operators. A particularly compact representation of the algebra associated with the manipulation of determinantal expansions is the method of second quantization or the occupation number representation . This is discussed in detail in several textbooks and review articles - - , to which the reader is referred for more detail. An especially entertaining presentation of second quantization is given by Mattuck . The usefulness of this approach is that it allows quite general algebraic manipulations to be performed on operator expressions. These formal manipulations are more cumbersome to perform in the wavefunction approach. It should be stressed, however, that these approaches are equivalent in content, if not in style, and lead to identical results and computational procedures. 

Given that the Hamiltonian is a one- and two-electron operator, basis set orthogonality and the restriction of Brillouin s fheorem, (4>hf H /s) = 0, introduce a formally and heuristically valuable selecfion rule for the MEP. It follows from Eqs. (4 and 5b) fhaf... 

What are these "integrals" to which we have referred From the fact that the Schrodinger Hamiltonian contains only one- and two-electron operators, it is straightforward to show [ 17] that most of the matrix elements [43] which arise in computing the SCF energy and its derivatives with respect to nuclear motion can be written in terms of integrals of the general form... 

The subscripts c and v denote core and valence, respectively. hy and gy are effective one- and two-electron operators, VCc represents the repulsion between all cores and nuclei of the system, and Vccp denotes the CPPs. The total number of electrons in the neutral system n and the number of valence electrons nv are related by the charges of the nuclei Z and the corresponding core charges Q ... 

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