Multi-configurational self-consistent field coefficients

Question

The total wave function of $H_2$ can be written as a linear combination of configurations built from bonding and anti-bonding orbitals:

$${\displaystyle \Psi _{\text{MC}}=C_{1}\Phi _{1}+C_{2}\Phi _{2},}$$

where $\Phi_2$ is the electronic configuration $(φ_2)^2$. In this multiconfigurational description of the $\ce{H2}$ chemical bond, $C_1$ = 1 and $C_2$ = 0 close to equilibrium, and $C_1$ will be comparable to $C_2$ for large separations.

How are the coefficients calculated?

Answer 1

In a MCSCF approach, the coefficients $C_I$ are calculated by minimizing the energy $\langle \Psi_{\mathrm{MC}}|H|\Psi_{\mathrm{MC}} \rangle$ with respect to (1) the MOs contributing to the configurations $\Phi_1$ and $\Phi_2$ (herein we have only two active orbitals, $\varphi_1$ and $\varphi_2$), and also (2) the coefficients $C_1$ and $C_2$, subject to (1) orthonormality conditions of the MOs and (2) orthonormality conditions of the configurations.

Although this uniquely determines the mathematical problem to be solved, the actual algorithm of solving this minimization problem is another matter. For example, the following techniques are all quite popular:

1. Optimize $C_I$ holding $\varphi_i$ fixed (which is equivalent to full CI within the active space), then optimize $\varphi_i$ holding $C_I$ fixed, and so on, until both converge. The latter optimization can proceed by using the gradient only, or using both the gradient and the Hessian.
2. Optimize both $C_I$ and $\varphi_i$ at the same time, using only the gradient of the energy $\langle \Psi_{\mathrm{MC}}|H|\Psi_{\mathrm{MC}} \rangle$ with respect to $C_I$ and $\varphi_i$.
3. Optimize both $C_I$ and $\varphi_i$ at the same time, using both gradient and Hessian information.

Implementing any such algorithm, especially when fast convergence is desired, is highly non-trivial. Anyway I hope the above answer already gives you a rough idea about how this works.