Born-Oppenheimer Hamiltonian operator

The Born-Oppenheimer Hamiltonian operator H = (T + V) for the fixed-nucleus approximation (Eq. 2.5) is expanded in powers of the displacement (R — Rg) near Rg and second-order perturbation theory is used to calculate E correct to second-order. Then k is given by... 

H being the Hamiltonian operator. Throughout this Chapter, we use the usual non-relativistic Born-Oppenheimer Hamiltonian in which no spin operators occur and the nuclear positions are considered fixed. 

In Eq. (3.2), p, fiv, fiz, fia are dipole moment operators and , E2, 3, E4 are the temporal field amplitudes corresponding to each of the four photons. Formally, H is the full electronic-nuclear Hamiltonian. However, in practice, H is the Born-Oppenheimer Hamiltonian of either the ground or excited electronic state, depending on the resonance condition satisfied by the preceding photons. 

Let 0> be the exact -electron ground state of the Born-Oppenheimer Hamiltonian H for a given atomic or molecular system. Likewise, let X > be some exact excited state of interest for the same system with the same nuclear geometry. The corresponding state energies are denoted and respectively. For excitation energy calculations X> is an excited N -electron state, whereas in ionization potential or electron affinity cases X > is an - l)-electron state or an (A -l- l)-electron state, respectively. The commutator of H with the operator Oj = X><0 is easily evaluated,... 

Taking the nonrelativistic Born-Oppenheimer Hamiltonian in which the nuclei are in fixed positions and in which there are no spin operators, we can write the exact stationary state wave function in the form [97]... 

This is a post Born-Oppenheimer scheme. It retains the essential idea of separability but the eleetronie base functions are diabatic functions. These functions are obtained from one Hamiltonian operator, namely the electronic operator defined in eq. (5). 

In the crude Born-Oppenheimer approximations, the oscillator strength of the 0-n vibronic transition is proportional to (FJ)2. Furthermore, the Franck-Condon factor is analytically calculated in the harmonic approximation. From the hamiltonian (2.15), it is clear that the exciton coupling to the field of vibrations finds its origin in the fact that we use the same vibration operators in the ground and the excited electronic states. By a new definition of the operators, it becomes possible to eliminate the terms B B(b + b ), BfB(b + hf)2. For that, we apply to the operators the following canonical transformation ... 

For the N-electron atom, we have seen (Section 3.7) several terms in the Hamiltonian operator. We collect here some more terms, to come to a "final list," within the Born-Oppenheimer approximation of a fixed nucleus ... 

It is also known, in the Born-Oppenheimer approximation, that the eigenfunctions of the full Hamiltonian operator (1) may be factorized into an electronic wave-function and a nuclear one ... 

Within the Born-Oppenheimer approximation, the number of electrons, N, and the external potential (from nonelectronic forces), v(r), completely determine the Hamiltonian operator, and, as a consequence, they also determine all the properties of the system. Specifically, the ground-state energy, E, is a concave function of the number of electrons and a convex functional of the external potential,... 

The Hamiltonian is now a pure nuclear operator dictating, in the Born-Oppenheimer approximation, the evolution of the nuclear wavefunction [115]. The electrons enter Hn only through a potential energy term, o(R/), added to the bare nuclei-nuclei interaction Vnn- This potential energy term due to the electrons is the ground state energy of the electronic system at fixed ionic configuration. 

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 ... 

In the clamped-nucleus Born-Oppenheimer approximation, with neglect of relativistic effects, the molecular Hamiltonian operator in atomic units takes the form... 

The Hamiltonian we adopt is a 2 x 2 matrix of operators. It represents the ground and the excited electronic states within the Born-Oppenheimer approximation, coupled by the radiation field interacting with the dipole operators ... 

Of paramount importance is the energy operator, called the Hamiltonian, which, for an /V-electron M-atom molecule treated within the Born-Oppenheimer approximation, may be written as... 

If these Born-Oppenheimer product wave functions are to approximate Hamiltonian eigenvectors, we have to minimize all off-diagonal matrix elements K L and u v). To this end, the electronic wave functions are chosen to be eigenvectors of a part of the Hamiltonian operator called the electronic Hamiltonian (adiabatic states) ... 

There are numerous interactions which are ignored by invoking the Born-Oppenheimer approximation, and these interactions can lead to terms that couple different adiabatic electronic states. The full Hamiltonian, H, for the molecule is the sum of the electronic Hamiltonian, the nuclear kinetic energy operator, Tf, the spin-orbit interaction, H, and all the remaining relativistic and hyperfine correction terms. The adiabatic Born-Oppenheimer approximation assumes that the wavefunctions of the system can be written in terms of a product of an electronic wavefunction, (r, R), a vibrational wavefunction, Xni( )> rotational wavefunction, and a spin wavefunction, Xspin- However, such a product wave-function is not an exact eigenfunction of the full Hamiltonian for the... 

The force -dE/dR, which acts on nucleus a of coordinates X Y, and Z ,is the negative gradient of the Born-Oppenheimer potential energy hypersurface E This force is equal to the Hellmann-Feynman force (F ).Differentiation of the Hamiltonian of equation (33) by R , leading to equation (32), eliminates all of the electronic kinetic energy terms and all of the terms describing electron-electron interactions. In fact, the force operator F is a one-electron operator, and the corresponding expectation value (F > has the actual form of... 

---