Electronic Hamiltonian

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The Electronic Hamiltonian is an operator in quantum mechanics (and in particular quantum chemistry) which describes the motions of electrons and nuclei in a polyatomic molecule. The terminology is sometimes used interchangeably to mean either the Electronic molecular Hamiltonian or the full electronic Hamiltonian. The latter includes a kinetic energy operator corresponding to the contributions from the nuclei.

There are a number of interrelated concepts associated with the term "Electronic Hamiltonian". These include the following:

Full electronic Hamiltonian
Electronic Hamiltonian
Nuclear Hamiltonian
Clamped Hamiltonian

Depending on the context, "full electronic Hamiltonian" may be used interchangeably with "electronic Hamiltonian". Furthermore, the "Clamped Hamiltonian" or may be used interchangeably with "Electronic Hamiltonian". The latter is usually used only when discussing various methods associated with the Born-Oppenheimer approximation.

Contents

1 Full electronic Hamiltonian

1.1 Electronic Hamiltonian and potential energy surfaces

2 Electronic molecular Hamiltonian
3 Adiabatic and diabatic states

Full electronic Hamiltonian

Let R denote the vector of nuclear coordinates, and r the vector of electronic coordinates.

The full electronic Hamiltonian consists of 5 terms. They are

The kinetic energy operators for each nucleus in the system;
The kinetic energy operators for each electron in the system;
The potential energy between the electrons and nuclei - the total electron-nucleus Coulombic attraction in the system;
The potential energy arising from Coulombic electron-electron repulsions
The potential energy arising from Coulombic nuclei-nuclei repulsions - also known as the nuclear repulsion energy. See electric potential for more details.

[\hat^n = - \sum_i \frac \frac]
[\hat^e = - \sum_i \frac \frac]
[\hat^ = - \sum_i \sum_j \frac]
[\hat^ = \sum_i \sum_ \frac]
[\hat^ = \sum_i \sum_ \frac ]

Hence, the full electronic Hamiltonian is

[\hat_ = \hat^n + \hat^e + \hat^ + \hat^ + \hat^]

Electronic Hamiltonian and potential energy surfaces

Very often, the electronic Hamiltonian is defined to be

[\hat^e = \hat^e + \hat^ + \hat^ + \hat^]

so that the full Hamiltonian would be written as

[\hat_ = \hat^e + \hat^n].

In this case, the kinetic energy operator [\hat^n] would be known as the nuclear Hamiltonian. The electronic Hamiltonian contains all three potential terms because their sum

[\hat \left ( \mathbf , \mathbf \right ) = \hat^ + \hat^ + \hat^ ]

is the expression which gives rise to the physically meaningful potential energy surfaces ubiquitous in chemistry, for fixed nuclear geometry [\mathbf].

Written in atomic units, the electronic Hamiltonian becomes:

[\hat H_=\sum_\nabla_i^2}-\sum_-\mathbf\right |}}}+\frac\sum_-\mathbf\right |}}}+\frac\sum_-\mathbf\right |}}}]

where

[\mathbf] is the vector position of electron [i] with vector components in Bohr radii,
[Z_a] is the charge of fixed nucleus a in units of the elementary charge,
[\mathbf] is the vector position of nucleus [a] with vector components in Bohr radii.

Electronic molecular Hamiltonian

The electronic molecular Hamiltonian is the term of the molecular Hamiltonian obtained when the molecular geometry is frozen. This is also known as the clamped Hamiltonian or clamped Hamiltonian approximation. Within the Born-Oppenheimer approximation, the electronic Hamiltonian is said to depend adiabatically on the molecular geometry. Its discrete eigenvalues are called potential energy surfaces and the corresponding eigenstates the electronic states of the molecule. The electronic states are labelled according to their group representation and spin multiplicity.

Adiabatic and diabatic states

By definition, the adiabatic states are diagonal in the electronic Hamiltonian. A consequence is that it is not diagonal in the kinetic energy operator. The off diagonal terms of this operator are known as the nonadiabatic operator.

Strictly diabatic states do not exist in general, although in the ideal case, it is diagonal in the kinetic energy operator, and off diagonal in the electronic Hamiltonian. In other words, the diabatic states minimize the magnitude of the contributions of the nonadiabatic operator.

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