# Multi-configurational self-consistent field coefficients

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?

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.
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.