The Schrödinger equation lies at the heart of quantum mechanics and underpins our understanding of molecular behavior. In this section, we explain how the time-independent Schrödinger equation (TISE) connects to real-world observations and how the Born-Oppenheimer approximation allows us to make tractable predictions in molecular systems.
The full time-independent Schrödinger equation (TISE) for a molecule is given by:
$$ \hat{H} \Psi(\mathbf{r}, \mathbf{R}) = E \Psi(\mathbf{r}, \mathbf{R}) $$
Where:
Under the Born–Oppenheimer approximation, the electronic Hamiltonian is:
$$ \hat{H} = -\sum_i \tfrac{1}{2}\nabla_i^2 -\sum_{iA} \tfrac{Z_A}{|\mathbf r_i-\mathbf R_A|} +\sum_{i<j} \tfrac{1}{|\mathbf r_i-\mathbf r_j|} +\sum_{A<B} \tfrac{Z_A Z_B}{|\mathbf R_A-\mathbf R_B|} $$
Vectors are bold; hats denote operators. The last term is the nuclear–nuclear repulsion (constant shift for fixed nuclei).
Because nuclei are much heavier and slower than electrons, we approximate the total wavefunction as:
$$ \Psi(\mathbf{r}, \mathbf{R}) \approx \psi_{\text{el}}(\mathbf{r}; \mathbf{R}) \chi(\mathbf{R}) $$
This allows us to solve first for the electronic part:
$$ \hat{H}_{\text{el}}(\mathbf{r}; \mathbf{R}) \psi_{\text{el}}(\mathbf{r}; \mathbf{R}) = E_{\text{el}}(\mathbf{R}) \psi_{\text{el}}(\mathbf{r}; \mathbf{R}) $$
and then solve for nuclear motion using $E_{\text{el}}(\mathbf{R})$ as the potential energy surface.
Electrons:
| Component | Description | Observed Phenomena |
|---|---|---|
| Schrödinger Equation | Governs total quantum behavior | Foundation for all predictions |
| Electrons | Fast, light, quantum adaptable | Color, bonding, redox, conductivity |
| Nuclei | Slow, heavy, define structure | Shape, IR spectra, isotope effects |
| Born-Oppenheimer Approximation | Separates electrons and nuclei | Enables tractable modeling |
Even when considering only the electronic Hamiltonian, finding an exact solution to the Schrödinger equation remains infeasible for systems with more than one electron. Fortunately, approximate solutions can be obtained using variational methods.
The variational principle states that the expectation value of the Hamiltonian, calculated with any trial wavefunction that is properly normalized and satisfies the correct boundary conditions, will always be greater than or equal to the true ground state energy. Symbolically:
$$ \langle \psi_{\text{trial}} | \hat{H}_{\text{el}} | \psi_{\text{trial}} \rangle \geq E_0 $$
Here, $E_0$ is the exact ground-state energy, and the wavefunction $\psi_{\text{trial}}$ becomes a variational parameter. In mathematical terms, the energy is a functional of the wavefunction, meaning small changes in the wavefunction lead to changes in the predicted energy.
By minimizing the expectation value of the Hamiltonian with respect to variations in the wavefunction, we can find an optimal wavefunction that best approximates the true ground state.
One of the most widely used approaches is the Hartree-Fock method, which applies the variational principle under the assumption that the total wavefunction can be approximated by a single Slater determinant. There are two main formulations:
To go beyond Hartree-Fock, methods such as Rayleigh-Schrödinger perturbation theory and configuration interaction (CI) are used. These incorporate electron correlation effects neglected by Hartree-Fock, leading to more accurate descriptions of molecular systems.