Total molecular Hamiltonian operator,

In section 2 we define the total molecular Hamiltonian operator describing both nuclear and electronic motion. The Hartree-Fock theory for nuclei and electrons is presented in section 3 and a many-body perturbation theory which uses this as a reference is developed in section 4. The diagrammatic perturbation theory of nuclei and... 

The total molecular Hamiltonian operator for a system containing N nuclei and n electrons may be written... 

We add the operator —p, ER to the total molecular Hamiltonian. According to Eq. (3.1), the electronic Hamiltonian of the molecule in the field due to the solvent is then He — p ER. The electronic Schrodinger equation is then solved using this modified Hamiltonian. This leads to a self-consistent solution where the electronic wave function and the electronic energy are modified due to the solvent field. Thus, polarization of the molecular electronic density (as described approximately above) is automatically included in this approach. 

In this scheme, the INT is partitioned into electrostatic, exchange, repulsion, polarization, and dispersion. Let us now look at the mathematical derivation of the various components. For a molecular system (AB) with wavefunction O and total energy Hamiltonian operator H, the expectation value is given as... 

The masses of the nuclei, m, are at least three orders of magnitude larger than the mass of an electron. We can therefore assume that the electrons will instantaneously adjust to a change in the positions of the nuclei and that we can find a wavefunction for the electrons for each arrangement of nuclei. In the Born-Oppenheimer approximation the total molecular Hamilton operator Hnuc,e from Eq. (2.1) is thus partitioned in the kinetic energy operator of the nuclei, field free electronic Hamiltonian defined as... 

For the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

Substituting the Slater determinant for the total molecular wavefunction T and inserting the explicit form of the Hamiltonian operator H into (5.14) gave the energy in terms of the spatial MO s 1jj, (Eq. 5.17) ... 

Hartree-Fock-Roothaan Closed-Shell Theory. Here [7], the molecular spin-orbitals it where the subscript labels the different MOs, are functions of (af, 2/", z") (where /z stands for the coordinate of the /zth electron) and a spin function. The configurational wave function is represented by a single determinantal antisymmetrized product wave function. The total Hamiltonian operator 2/F is defined by... 

Until this point, the consideration of electron-electron repulsion terms has been neglected in the molecular Hamiltonian. Of course, an accurate molecular Hamiltonian must account for these forces, even though an explicit term of this type renders exact solution of the Schrddinger equation impossible. The way around this obstacle is the same Hartree-Fock technique that is used for the solution of the Schrddinger equation in many-electron atoms. A Hamiltonian is constructed in which an effective potential of the other electrons substitutes for a true electron-electron reg sion term. The new operator is called the Lock operator, F. The orbital approximation is still used so that F can be separated into i (the total number of electrons) one-electron operators, Fi (19). 

The Schrodinger equation is the starting point for molecular problems. The symbol H is a differential operator called the Hamiltonian operator, which is analogous to the classical Hamiltonian, in as much as it is a sum of kinetic and potential energy terms. E is the total energy for the system. The wavefunction P depends on the position of all the particles comprising the system. proposed that I Fp, and not P,... 

We begin with some general considerations of perhaps lesser-known, but important, features of exact electronic wavefunctions. Our motive is to establish a theoretical framework together with a reasonably consistent notation in order to carry through the spin-coupled VB and other expansions of the total wavefunction. We consider an atomic or molecular system consisting of N electrons and A nuclei. We assume the Born-Oppenheimer separation and write the Hamiltonian operator for the motion of the electrons in the form ... 

The Hartree-Fock Hamiltonian operator can be obtained by the procedure in which the total energy of the system is varied to be minimum by an infinitesimal change of each molecular orbital. See Ref. 10. 

Bound states of the discrete spectrum of the atomic (molecular) Hamiltonian, Ha M, in which case E is the total energy, which is one of the real eigenvalues of the TISE, say E . (Bold letters symbolize operators.)... 

Many molecular hamiltonians commute with the total spin angular momentum operator, a fact that leads to the consideration of transformation properties of electron field operators under rotations in spin space. Basis functions, natural for such studies, are... 

This operator and consequently the effective Hamiltonian are complex and have an imaginary (antihermitian) part that describes damping of the molecular excited states by interaction with the radiation field. Rather than using the eigenfunctions (3) of H m, it is then appropriate to choose the regular eigenfunctions of the total effective Hamiltonian (16) to represent the molecular excitations (Sternheim and Walker, 1972). 

The occurrence of electron degeneracy brings two effects. First, the molecular Hamiltonian is invariant with respect to arbitrary symmetry operations, so that the adiabatic potential belongs to the totally symmetric representation (A-type or 2-type). Therefore, only certain combinations of QjQj or QjQjQk are allowed for symmetry coordinates and they must span the totally symmetric representation. Secondly, the interaction matrix U... 

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