Electronic Schrödinger Equation

The electronic Schrödinger equation is the fundamental equation of quantum chemistry. Every electronic structure method—including Hartree–Fock (HF), Density Functional Theory (DFT), Møller–Plesset Perturbation Theory (MP2), and Coupled Cluster (CC)—begins with this equation.

Its objective is straightforward:

Given the positions of the atomic nuclei, determine the electronic wavefunction and the corresponding energy of the molecule.

Unfortunately, for molecules containing more than one electron, the equation cannot be solved exactly. The remainder of this tutorial explains how increasingly sophisticated approximations ultimately lead to the Hartree–Fock method.

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The Time-Independent Schrödinger Equation

For a stationary quantum system, the time-independent Schrödinger equation is

\[ \hat{H}\Psi = E\Psi \]

where

Symbol	Meaning
\(\hat{H}\)	Hamiltonian operator
\(\Psi\)	Wavefunction
\(E\)	Total energy of the system

The Hamiltonian contains all the information about the physical system. Solving this equation provides both

• the allowed energies of the system, and
• the corresponding wavefunctions.

Note

The wavefunction itself cannot be measured directly. However, its square,

\[ |\Psi|^2 \]

represents the probability density of finding the electrons in space.

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What is the Hamiltonian?

The Hamiltonian represents the total energy operator.

For any molecular system, it consists of

• kinetic energy,
• potential energy.

In classical mechanics,

\[ E = T + V \]

where

• \(T\) is the kinetic energy,
• \(V\) is the potential energy.

Quantum mechanics follows the same principle, except that these quantities become operators.

Thus,

\[ \boxed{ \hat H = \hat T + \hat V } \]

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The Molecular Hamiltonian

A molecule contains

• nuclei,
• electrons.

Both possess kinetic energy, while several electrostatic interactions contribute to the potential energy.

Therefore,

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

where

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

This expression represents the exact non-relativistic Hamiltonian of a molecule.

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The Individual Energy Terms

Each contribution has a clear physical interpretation.

Nuclear Kinetic Energy

Since nuclei possess mass and move through space,

\[ \hat T_n = -\sum_A \frac{\hbar^2}{2M_A} \nabla_A^2 \]

where

• \(M_A\) is the mass of nucleus \(A\),
• \(\nabla_A^2\) is the Laplacian with respect to its coordinates.

Because nuclei are much heavier than electrons, this term often changes much more slowly.

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Electronic Kinetic Energy

Electrons also possess kinetic energy.

\[ \hat T_e = - \sum_i \frac{\hbar^2}{2m_e} \nabla_i^2 \]

where

• \(m_e\) is the electron mass,
• \(i\) labels each electron.

This operator governs the motion of the electrons throughout the molecule.

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Electron–Nucleus Attraction

Every electron is attracted to every nucleus through Coulomb's law.

\[ \hat V_{ne} = - \sum_{A} \sum_i \frac{Z_Ae^2} {4\pi\varepsilon_0r_{Ai}} \]

where

• \(Z_A\) is the atomic number,
• \(r_{Ai}\) is the distance between electron \(i\) and nucleus \(A\).

This negative contribution is responsible for binding electrons to atoms.

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Electron–Electron Repulsion

Since electrons have identical negative charges, they repel one another.

\[ \hat V_{ee} = \sum_{i<j} \frac{e^2} {4\pi\varepsilon_0r_{ij}} \]

where

\(r_{ij}\)

is the distance between electrons.

This is the most difficult term in quantum chemistry because every electron interacts with every other electron.

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Nuclear Repulsion

Similarly, the positively charged nuclei repel one another.

\[ \hat V_{nn} = \sum_{A<B} \frac{Z_AZ_Be^2} {4\pi\varepsilon_0R_{AB}} \]

where

\(R_{AB}\)

is the distance between nuclei.

For a fixed molecular geometry, this term remains constant.

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Why Can't We Solve This Equation Exactly?

For a hydrogen atom,

there is only one electron.

Since no electron–electron interaction exists,

the Schrödinger equation has an exact analytical solution.

However, molecules contain many electrons.

For a system containing \(N\) electrons,

each electron interacts with every other electron.

The wavefunction therefore depends on the coordinates of all electrons simultaneously,

\[ \Psi = \Psi (\mathbf r_1, \mathbf r_2, \dots, \mathbf r_N) \]

making the problem enormously complicated.

The electron–electron repulsion term couples the motion of every electron to every other electron, preventing an exact solution.

Warning

The electron–electron repulsion term is the primary reason why exact solutions are impossible for most molecules.

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The Need for Approximation

Because the exact Schrödinger equation cannot be solved for many-electron systems, quantum chemistry relies on carefully chosen approximations.

The first approximation is the

Born–Oppenheimer Approximation,

which separates nuclear motion from electronic motion.

This simplification transforms the full molecular Schrödinger equation into the electronic Schrödinger equation, which forms the starting point for Hartree–Fock theory.

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Key Takeaways

• The Schrödinger equation is the foundation of electronic structure theory.
• The Hamiltonian contains all kinetic and potential energy contributions.
• Electron–electron repulsion prevents exact analytical solutions for molecules.
• Most quantum chemistry methods begin by introducing approximations to simplify this problem.
• The first and most important approximation is the Born–Oppenheimer Approximation, discussed in the next chapter.

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