Electronic Schrödinger Equation¶
After applying the Born–Oppenheimer approximation, the motion of the nuclei is separated from that of the electrons. The remaining task is to solve the electronic Schrödinger equation, which determines the electronic wavefunction and corresponding electronic energy for a fixed arrangement of nuclei.
Every electronic structure method implemented in Gaussian—whether Hartree–Fock (HF), Density Functional Theory (DFT), Møller–Plesset perturbation theory (MP2), or Coupled Cluster (CC)—ultimately attempts to solve this equation, either exactly or approximately.
The electronic Schrödinger equation therefore forms the mathematical foundation of all quantum chemical calculations.
From the Molecular Problem to the Electronic Problem¶
The complete molecular Schrödinger equation describes both the electrons and nuclei simultaneously,
where
- \(\mathbf{r}\) represents the electronic coordinates,
- \(\mathbf{R}\) represents the nuclear coordinates,
- \(\hat{H}\) is the molecular Hamiltonian.
After invoking the Born–Oppenheimer approximation, the nuclear positions are treated as fixed parameters. The problem then reduces to solving
where
- \(\hat{H}_e\) is the electronic Hamiltonian,
- \(\psi_e\) is the electronic wavefunction,
- \(E_e\) is the electronic energy.
This is the equation solved during every Single Point Energy calculation.
What Does the Equation Mean?¶
The electronic Schrödinger equation asks a simple but profound question:
"How are the electrons distributed around a fixed arrangement of nuclei, and what is the corresponding electronic energy?"
The solution provides
- the molecular orbitals,
- the electron density,
- the electronic energy,
- and many observable molecular properties.
Once the electronic wavefunction has been obtained, numerous molecular properties can be calculated without performing additional quantum mechanical calculations.
Components of the Equation¶
The electronic Schrödinger equation contains three important quantities.
Electronic Hamiltonian¶
The Hamiltonian operator represents the total electronic energy of the system.
It includes
- kinetic energy of the electrons,
- attraction between electrons and nuclei,
- repulsion between electrons.
The Hamiltonian determines how electrons move within the electrostatic field created by the fixed nuclei.
Electronic Wavefunction¶
The wavefunction
contains all available information about the electronic state of the molecule.
Although the wavefunction itself has no direct physical meaning, measurable quantities such as
- electron density,
- dipole moments,
- molecular orbital energies,
- and atomic charges
can all be derived from it.
Finding the correct wavefunction is therefore the primary objective of every electronic structure calculation.
Electronic Energy¶
The quantity
is the electronic energy associated with the wavefunction.
For a Single Point Energy calculation, this is the value reported by Gaussian after SCF convergence.
Typical output appears as
where the energy is expressed in Hartree (atomic units).
Why Can't We Solve It Exactly?¶
For a molecule containing multiple electrons, each electron interacts with
- every nucleus,
- and every other electron.
The electron–electron repulsion term couples the motion of all electrons together.
As a result,
- the electronic coordinates cannot be separated,
- no analytical solution exists,
- and exact solutions are available only for very small systems such as the hydrogen atom.
For all chemically interesting molecules, approximate methods are required.
Approximate Methods¶
Different quantum chemical methods differ primarily in how they approximate the solution of the electronic Schrödinger equation.
| Method | Approximation |
|---|---|
| Hartree–Fock | Average electron–electron interaction |
| DFT | Uses electron density instead of the many-electron wavefunction |
| MP2 | Perturbative correction to Hartree–Fock |
| CCSD | Systematic treatment of electron correlation |
| CASSCF | Multiconfigurational wavefunction |
Although the mathematical formulations differ, all of these methods ultimately seek the lowest-energy electronic state consistent with the chosen approximation.
Connection with Gaussian¶
During a Single Point Energy calculation, Gaussian performs the following sequence:
Read Molecular Geometry
│
▼
Construct Electronic Hamiltonian
│
▼
Choose Computational Method
│
▼
Solve Electronic Schrödinger Equation
│
▼
Obtain Electronic Energy
The exact procedure used to solve the equation depends on the selected electronic structure method.
For this tutorial, the selected method is B3LYP, a Density Functional Theory (DFT) method.
Why Do We Need More Mathematics?¶
Although the electronic Schrödinger equation appears compact,
both the Hamiltonian and the wavefunction are mathematically complex.
Before discussing Hartree–Fock or Density Functional Theory, we must first understand the individual components of the electronic Hamiltonian.
This leads naturally to the next topic: the Molecular Hamiltonian, where each energy contribution—kinetic energy, electron–nucleus attraction, electron–electron repulsion, and nucleus–nucleus repulsion—is introduced and interpreted.
Summary¶
The electronic Schrödinger equation describes the motion of electrons in the electrostatic field generated by fixed nuclei. Solving this equation yields the electronic wavefunction and the corresponding electronic energy, which form the basis for nearly all molecular properties computed by Gaussian. Because exact solutions are impossible for multi-electron systems, practical quantum chemistry relies on approximate methods such as Hartree–Fock and Density Functional Theory, both of which build upon this fundamental equation.
Next Section¶
The next chapter examines the Molecular Hamiltonian, the operator that contains all the energy contributions governing the behavior of electrons in a molecule.