Complete Single Point Energy Algorithm¶

The previous chapters introduced the theoretical foundations of a Density Functional Theory calculation, including the Born–Oppenheimer approximation, Hartree–Fock theory, the Kohn–Sham formulation of Density Functional Theory, basis sets, the Self-Consistent Field (SCF) procedure, and the B3LYP exchange–correlation functional.

This chapter combines all of these concepts into a single computational workflow, illustrating how Gaussian converts an input file into the final electronic energy reported in the output.

Although many sophisticated numerical algorithms operate internally, the overall procedure follows a well-defined sequence of steps.

Step 1 — Read the Input File¶

Gaussian first reads the entire input file.

For example,

%nprocshared=1
%mem=2GB
%chk=formicacid.chk

# B3LYP/6-31G

0 1

Coordinates...

From this input Gaussian determines

computational method (B3LYP)
basis set (6-31G)
molecular geometry
charge
spin multiplicity
available memory
number of processors

At this stage no quantum mechanical calculations have yet been performed.

Step 2 — Apply the Born–Oppenheimer Approximation¶

The nuclei are assumed to remain fixed while solving the electronic problem.

Only the electronic Schrödinger equation is considered

\[ \hat H_{elec}\Psi = E\Psi \]

The nuclear positions supplied in the input become fixed parameters throughout the SCF calculation.

Step 3 — Construct Gaussian Basis Functions¶

Next,

Gaussian reads the chosen basis set

6-31G

and places the appropriate contracted Gaussian basis functions on every atom.

Instead of solving directly for molecular orbitals,

each orbital is written as

\[ \psi_i = \sum_\mu C_{\mu i} \chi_\mu \]

where

\(\chi_\mu\) are Gaussian basis functions,
\(C_{\mu i}\) are unknown coefficients.

The electronic problem has now become a matrix problem.

Step 4 — Build the One-Electron Integrals¶

Using the basis functions,

Gaussian evaluates

overlap integrals

\[ S_{\mu\nu} \]

kinetic-energy integrals

\[ T_{\mu\nu} \]

electron–nuclear attraction integrals

\[ V_{\mu\nu} \]

These quantities form the core Hamiltonian

\[ H_{core} = T + V. \]

These matrices remain fixed throughout the calculation because the molecular geometry does not change.

Step 5 — Generate an Initial Electron Density¶

Since the exact electron density is unknown,

Gaussian constructs an initial approximation using methods such as

Superposition of Atomic Densities (SAD)
Core Hamiltonian Guess
Previous checkpoint files

This initial density serves as the starting point for the SCF procedure.

Step 6 — Construct the Kohn–Sham Matrix¶

Using the current electron density,

Gaussian evaluates

Coulomb interaction
exchange interaction
correlation contribution

through the B3LYP functional.

The Kohn–Sham matrix is then assembled,

\[ F_{KS} = H_{core} + J + V_{XC}. \]

Unlike the core Hamiltonian,

this matrix changes during every SCF iteration because it depends on the electron density.

Step 7 — Solve the Kohn–Sham Equations¶

The matrix equation

\[ FC = SC\varepsilon \]

is solved to obtain

molecular orbital coefficients,
molecular orbital energies.

The occupied molecular orbitals are then used to construct a new electron density.

Step 8 — Perform the Self-Consistent Field Cycle¶

The updated electron density is compared with the previous density.

If convergence has not been achieved,

Gaussian repeats

Density

↓

Kohn–Sham Matrix

↓

Diagonalization

↓

New Orbitals

↓

New Density

until the electron density no longer changes within the specified convergence thresholds.

Step 9 — SCF Convergence¶

When convergence is reached,

Gaussian prints

SCF Done:

E(RB3LYP)

=
-379.119248 Hartree

This is the final electronic energy corresponding to the converged electron density.

It represents the electronic energy for the fixed nuclear geometry supplied in the input file.

Step 10 — Save the Wavefunction¶

Finally,

Gaussian writes several files,

including

checkpoint file (.chk)
formatted checkpoint (.fchk, if requested)
output file (.log or .out)

These files contain

molecular orbitals,
electron density,
orbital energies,
Mulliken populations,
dipole moments,
and other calculated properties.

Subsequent calculations such as geometry optimization, frequency analysis, NMR, or excited-state calculations can reuse the checkpoint file.

Overall Workflow¶

The complete procedure can be summarized as

Read Input File
        │
        ▼
Read Molecular Geometry
        │
        ▼
Apply Born–Oppenheimer Approximation
        │
        ▼
Generate Gaussian Basis Functions
        │
        ▼
Construct One-Electron Integrals
        │
        ▼
Generate Initial Electron Density
        │
        ▼
Build Kohn–Sham Matrix (B3LYP)
        │
        ▼
Solve Roothaan–Hall / Kohn–Sham Equations
        │
        ▼
Construct New Electron Density
        │
        ▼
SCF Convergence?
     │         │
     │ No      │ Yes
     ▼         ▼
Repeat      Final Energy
                 │
                 ▼
Write Output Files

Summary¶

A Gaussian single-point energy calculation is a sequence of well-defined computational steps. Starting from the molecular geometry and chosen basis set, Gaussian constructs basis functions, evaluates the necessary one-electron and electron–electron terms, solves the Kohn–Sham equations through an iterative Self-Consistent Field procedure, and produces the final electronic energy once convergence is achieved. Although the underlying mathematics is sophisticated, the overall workflow follows a logical progression from molecular structure to electronic energy.

Conclusion¶

You now have the theoretical background needed to understand how Gaussian performs a Single Point Energy calculation. The following chapters move beyond theory and describe additional calculation types, including Geometry Optimization, Frequency Analysis, Optimization + Frequency, IRC, Potential Energy Surface Scans, NMR, and Wavefunction Stability Analysis.