LESSON NOTES · 01

Separate nuclei; antisymmetrize electrons

Prerequisites: Variational principle; products and determinants

Prerequisite lessons: Variational principle and finite bases

Conventions

Use fixed nuclear coordinates \(R\), electronic coordinates \(x=(r,\sigma)\) with spin summation, and atomic units: energy in hartree, length in bohr. Molecular orbitals are one-electron functions used to build a many-electron state, not labeled electrons’ trajectories.

1. Separate electronic and nuclear terms

Start with the nonrelativistic Coulomb Hamiltonian. Solve the electronic problem at each chosen nuclear geometry; electronic energy plus nuclear repulsion defines a potential-energy surface. Expand the full state in geometry-dependent electronic eigenstates.

\[ \begin{aligned}H&=T_N+H_e(R)+V_{NN}(R),\\H_e&=\sum_i\left[-\tfrac12\nabla_i^2-\sum_A\frac{Z_A}{|r_i-R_A|}\right]+\sum_{i<j}\frac1{r_{ij}},\\H_e(R)\psi_k(x;R)&=E_{e,k}(R)\psi_k(x;R),\quad\Psi(x,R)=\sum_k\chi_k(R)\psi_k(x;R).\end{aligned} \]

2. See exactly what Born–Oppenheimer drops

Nuclear kinetic energy differentiates both factors. The product rule produces derivative couplings; dropping them at leading order gives a single-surface nuclear equation. Heavy nuclei motivate the approximation, but a small electronic gap can make those couplings important. A crossing is not by itself a universal failure criterion.

\[ \nabla_R^2(\chi\psi)=\psi\nabla_R^2\chi+2(\nabla_R\chi)\cdot(\nabla_R\psi)+\chi\nabla_R^2\psi,\quad[T_N+E_{e,k}+V_{NN}]\chi_k=E\chi_k. \]

3. Antisymmetrize complete electron coordinates

Electrons are identical fermions. Exchanging coordinates including spin must change the many-electron sign. For orthonormal spin orbitals, a normalized determinant enforces this: swapping two electron columns changes sign; duplicate occupied orbital rows give zero.

\[ \Phi(x_1,\ldots,x_N)=\frac1{\sqrt{N!}}\det[\varphi_p(x_q)]. \]

Separate nuclei; antisymmetrize electrons — schematic under the stated model assumptions

Teaching schematic drawn from the equations or algorithm steps in this lesson; not measured data.

Worked example: shared spatial orbital

Two electrons can occupy one spatial orbital with opposite spin because the spin orbitals differ. The spatial part is symmetric and the singlet spin part antisymmetric, so their product has the required total antisymmetry.

\[ \Phi=\phi(r_1)\phi(r_2)\frac{\alpha(1)\beta(2)-\beta(1)\alpha(2)}{\sqrt2}. \]

Check yourself and limits

Is nuclear repulsion omitted from molecular total energy?

Answer and reasoning

No; \(V_{NN}\) is necessary even if constant at one geometry.

Does a determinant capture arbitrary electron correlation?

Answer and reasoning

No; it restricts the many-electron ansatz.

Does heavier nuclear mass guarantee BO validity?

Answer and reasoning

No; the electronic gap and derivative couplings also matter.

References and further reading

Derivations and numerical examples here are original teaching constructions, not copied passages or reported research data.

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