Born-Oppenheimer approximation Hamiltonian

In currently available software, the Hamiltonian above is nearly never used. The problem can be simplified by separating the nuclear and electron motions. This is called the Born-Oppenheimer approximation. The Hamiltonian for a molecule with stationary nuclei is... 

Since nuclei are much heavier than electrons and move slower, the Born-Oppenheimer Approximation suggests that nuclei are stationary and thus that we can solve for the motion of electrons only. This leads to the concept of an electronic Hamiltonian, describing the motion of electrons in the potential of fixed nuclei. 

The Born-Oppenheimer approximation allows the two parts of the problem to be solved independently, so we can construct an electronic Hamiltonian which neglects the kinetic energy term for the nuclei ... 

The Hamiltonian for this system should include the kinetic and potential energy of the electron and both of the nuclei. However, since the electron mass is more than a thousand times smaller than that of the lightest nucleus, one can consider the nuclei to be effectively motionless relative to the quickly moving electron. This assumption, which is basically the Born-Oppenheimer approximation, allows one to write the Schroedinger equation neglecting the nuclear kinetic energy. For the Hj ion the Born-Oppenheimer Hamiltonian is... 

The transition from (1) and (2) to (5) is reversible each implies the other if the variations 5l> admitted are completely arbitrary. More important from the point of view of approximation methods, Eq. (1) and (2) remain valid when the variations 6 in a trial function are constrained in some systematic way whereas the solution of (5) subject to model or numerical approximations is technically much more difficult to handle. By model approximation we shall mean an approximation to the form of as opposed to numerical approximations which are made at a lower level once a model approximation has been made. That is, we assume that H, the molecular Hamiltonian is fixed (non-relativistic, Born-Oppenheimer approximation which itself is a model in a wider sense) and we make models of the large scale electronic structure by choice of the form of and then compute the detailed charge distributions, energetics etc. within that model. 

Also note that Eeiec is the potential energy for nuclear motion in the Born-Oppenheimer approximation. Thus the classical Hamiltonian H for nuclear (vibrational) motion is given by... 

Within the Born-Oppenheimer approximation, the non-relativistic electronic Hamiltonian of an A-electron molecular system in the presence of an external potential can be written (in atomic units) as... 

The electronic Hamiltonian within the Born-Oppenheimer approximation may be split into two parts... 

Within the Born-Oppenheimer approximation, we assume the nuclei are held fixed while the electrons move really fast around them, (note Mp/Me 1840.) In this case, nuclear motion and electronic motion are seperated. The last two terms can be removed from the total hamiltonian to give the electronic hamiltonian, He, since Vnn = K, and = 0. The nuclear motion is handled in a rotational/vibrational analysis. We will be working within the B-0 approximation, so realizing... 

In this case the zero-order electronic wave functions are, in principle, referred to a Hamiltonian that contains the potential from the ions at their actual positions, i.e., the electrons follow the ionic motion adiabatically. Since both these approximations are sometimes referred to as the Born Oppenheimer approximation, this has led to confusion in terminology for example, Mott (1977) refers to the Born-Oppenheimer approximation, but gives wave functions of the adiabatic type, whereas Englman (1972) differentiates between the two forms, but specifically calls the static form the Bom Oppenheimer method. [We note that, historically, the adiabatic form was first suggested by Seitz (1940)—see, for example, Markham (1956) or Haug and Sauermann (1958)]. In this chapter, we shall preferentially use the terminology static and adiabatic. [Note that the term crude adiabatic is also sometimes used for the static approximation, mainly in the chemical literature—see, for example, Englman (1972, 1979).]... 

Up to now, we have been discussing many-particle molecular systems entirely in the abstract. In fact, accurate wave functions for such systems are extremely difficult to express because of the correlated motions of particles. That is, the Hamiltonian in Eq. (4.3) contains pairwise attraction and repulsion tenns, implying that no particle is moving independently of all of the others (the term correlation is used to describe this interdependency). In order to simplify the problem somewhat, we may invoke the so-called Born-Oppenheimer approximation. This approximation is described with more rigor in Section 15.5, but at this point we present the conceptual aspects without delving deeply into the mathematical details. 

INTRODUCTION. A standard and universal description of various nonlinear spectroscopic techniques can be given in terms of the optical response functions (RFs) [1], These functions allow one to perturbatively calculate the nonlinear response of a material system to external time-dependent fields. Normally, one assumes that the Born-Oppenheimer approximation is adequate and it is sufficient to consider the ground and a certain excited electronic state of the system, which are coupled via the laser fields. One then can model the ground and excited state Hamiltonians via a collection of vibrational modes, which are usually assumed to be harmonic. The conventional damped oscillator is thus the standard model in this case [1]. 

The Born- Oppenheimer approximation. With spin-orbit and other relativistic interactions omitted, the Hamiltonian of a polyatomic molecule is... 

We now consider the nuclear motions of polyatomic molecules. We are using the Born-Oppenheimer approximation, writing the Hamiltonian HN for nuclear motion as the sum of the nuclear kinetic-energy TN and a potential-energy term V derived from solving the electronic Schrodinger equation. We then solve the nuclear Schrodinger equation... 

Point-group symmetry exists only within a particular Born-Oppen-heimer approximation. Though point-group symmetry often has little to do with spin conservation, it will be found in Section VIII that spin concepts and point-group symmetry are intermingled when a Hamiltonian involving spin interactions is considered. Also, we will find that Born-Oppenheimer approximations are important in Franck-Condon factors Franck-Condon factors are, in turn, critical in determining transition probabilities for a number of spin-forbidden processes. 

The Born-Oppenheimer approximation permits the molecular Hamiltonian H to be separated into a component H, that depends only on the coordinates of the electrons relative to the nuclei, plus a component depending upon the nuclear coordinates. This in turn can be wriuen as a sum Hr + H, of terms for vibrational and rotational motion of ihe nuclei. 

The material model consists of a large assembly of molecules, each well characterized and interacting according to the theory of noncovalent molecular interactions. Within this framework, no dissociation processes, such as those inherently present in water, nor other covalent processes are considered. This material model may be described at different mathematical levels. We start by considering a full quantum mechanical (QM) description in the Born-Oppenheimer approximation and limited to the electronic ground state. The Hamiltonian in the interaction form may be written as ... 

Whereas the quantum-mechanical molecular Hamiltonian is indeed spherically symmetrical, a simplified virial theorem should apply at the molecular level. However, when applied under the Born-Oppenheimer approximation, which assumes a rigid non-spherical nuclear framework, the virial theorem has no validity at all. No amount of correction factors can overcome this problem. All efforts to analyze the stability of classically structured molecules in terms of cleverly modified virial schemes are a waste of time. This stipulation embraces the bulk of modern bonding theories. 

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