Molecular Hamiltonian¶

The electronic Schrödinger equation introduced in the previous chapter,

\[ \hat{H}_e \psi_e = E_e \psi_e, \]

contains a single operator that governs the entire behavior of the electrons: the electronic Hamiltonian.

The Hamiltonian represents the total energy of the electronic system. It combines all the physical interactions experienced by the electrons in a molecule and determines both the electronic wavefunction and the corresponding electronic energy.

Every electronic structure method implemented in Gaussian, from Hartree–Fock (HF) and Density Functional Theory (DFT) to more advanced post-Hartree–Fock methods, begins by constructing and solving this Hamiltonian.

What is a Hamiltonian?¶

In quantum mechanics, every observable physical quantity is represented by an operator.

The Hamiltonian operator,

\[ \hat H, \]

is the operator corresponding to the total energy of the system.

Applying the Hamiltonian to the wavefunction yields the allowed energy levels,

\[ \hat H \Psi = E \Psi. \]

Thus, solving the Schrödinger equation is essentially equivalent to finding the eigenvalues and eigenfunctions of the Hamiltonian.

Molecular Hamiltonian¶

For a molecule containing many electrons and nuclei, the complete molecular Hamiltonian is

\[ \boxed{ \hat H = \hat T_e + \hat T_n + \hat V_{ee} + \hat V_{nn} + \hat V_{en} } \]

where

Symbol	Meaning
\(\hat T_e\)	Electronic kinetic energy
\(\hat T_n\)	Nuclear kinetic energy
\(\hat V_{ee}\)	Electron–electron repulsion
\(\hat V_{nn}\)	Nucleus–nucleus repulsion
\(\hat V_{en}\)	Electron–nucleus attraction

These five terms together describe every interaction present in a molecule.

After the Born–Oppenheimer Approximation¶

Since Gaussian assumes fixed nuclei during a Single Point Energy calculation, the nuclear kinetic energy is removed from the problem.

The remaining electronic Hamiltonian becomes

\[ \boxed{ \hat H_e = \hat T_e + \hat V_{en} + \hat V_{ee} + V_{nn} } \]

Notice that

the nuclei no longer move,
the nucleus–nucleus repulsion becomes a constant,
only the electrons remain quantum mechanical particles.

This simplified Hamiltonian is solved during every electronic structure calculation.

Electronic Kinetic Energy¶

The first contribution is the kinetic energy of the electrons,

\[ \boxed{ \hat T_e = - \frac12 \sum_i \nabla_i^2 } \]

where

the summation runs over all electrons,
\(\nabla_i^2\) is the Laplacian operator acting on electron \(i\).

This term describes the motion of electrons throughout the molecule.

Without kinetic energy, electrons would simply collapse onto the nuclei.

Electron–Nucleus Attraction¶

Electrons are attracted to the positively charged nuclei through the Coulomb interaction.

The corresponding operator is

\[ \boxed{ \hat V_{en} = - \sum_i \sum_A \frac{Z_A}{r_{iA}} } \]

where

\(Z_A\) is the nuclear charge,
\(r_{iA}\) is the distance between electron \(i\) and nucleus \(A\).

The negative sign reflects the attractive nature of this interaction.

This term is responsible for binding electrons to atoms and forming molecules.

Electron–Electron Repulsion¶

Every electron repels every other electron.

The interaction is described by

\[ \boxed{ \hat V_{ee} = \sum_{i<j} \frac1{r_{ij}} } \]

where

\(r_{ij}\) is the distance between electrons \(i\) and \(j\).

This is the most difficult term in the Hamiltonian because the motion of every electron depends on the positions of all the others.

As a consequence,

the electronic Schrödinger equation cannot be solved exactly,
and approximate methods such as Hartree–Fock and DFT become necessary.

Nucleus–Nucleus Repulsion¶

The positively charged nuclei also repel one another.

Their interaction is

\[ \boxed{ V_{nn} = \sum_{A<B} \frac{Z_A Z_B}{R_{AB}} } \]

where

\(R_{AB}\) is the distance between nuclei \(A\) and \(B\).

Since the nuclei are fixed during a Single Point Energy calculation, this quantity remains constant throughout the calculation.

Although constant, it contributes to the total molecular energy reported by Gaussian.

The Electronic Hamiltonian in Gaussian¶

Combining all contributions gives

\[ \boxed{ \hat H_e = - \frac12 \sum_i \nabla_i^2 - \sum_i \sum_A \frac{Z_A}{r_{iA}} + \sum_{i<j} \frac1{r_{ij}} + V_{nn} } \]

This equation is the starting point for

Hartree–Fock theory,
Density Functional Theory,
MP2,
Coupled Cluster,
and most other quantum chemical methods.

The difference between these methods lies not in the Hamiltonian itself, but in how the electronic wavefunction (or electron density) is approximated.

Physical Interpretation¶

Each term has a clear physical meaning.

Electronic Hamiltonian

        │
        ├── Electron kinetic energy
        │
        ├── Electron–nucleus attraction
        │
        ├── Electron–electron repulsion
        │
        └── Nucleus–nucleus repulsion

The final electronic energy obtained from Gaussian represents the balance between these competing interactions.

For example,

kinetic energy tends to delocalize electrons,
electron–nucleus attraction localizes electrons around nuclei,
electron–electron repulsion pushes electrons apart,
nucleus–nucleus repulsion increases as atoms approach one another.

The observed molecular geometry corresponds to a balance between all of these energy contributions.

Why is the Electron–Electron Term Difficult?¶

Among all terms in the Hamiltonian, the electron–electron repulsion term presents the greatest computational challenge.

Each electron moves in the electric field created by every other electron, meaning that the electronic motions are coupled.

Because of this coupling,

the electronic wavefunction cannot be separated into independent one-electron equations,
and no exact analytical solution exists for multi-electron molecules.

The remainder of modern quantum chemistry is largely devoted to developing efficient approximations for treating this interaction.

Summary¶

The molecular Hamiltonian is the total energy operator of a molecular system. After applying the Born–Oppenheimer approximation, it consists of four principal contributions: electronic kinetic energy, electron–nucleus attraction, electron–electron repulsion, and nucleus–nucleus repulsion. The electron–electron interaction makes the many-electron Schrödinger equation impossible to solve exactly, motivating the development of approximate methods such as Hartree–Fock and Density Functional Theory. Every electronic structure calculation in Gaussian begins by constructing this Hamiltonian.

Next Section¶

The next chapter introduces the Hartree Product, the first approximation used to simplify the many-electron wavefunction before developing the more accurate Slater determinant and Hartree–Fock theory.