Approximations to the full CCD equations
To solve Eq.
(2), we would thus start with a guess for the unknown amplitudes, typically using the wave operator defined by first order in perturbation theory, leading to a zeroth approximation to the energy given by second-order perturbation theory for the correlation energy.
A simple approach to the solution of Eq.
(2), is to thus to
- Start with a guess for the amplitudes and compute the zeroth approximation to the correlation energy
- Use the ansatz for the amplitudes to solve Eq. (2) via for example your root-finding method of choice (Newton's method or modifications thereof can be used) and continue these iterations till the correlation energy does not change more than a prefixed quantity \( \lambda \); \( \Delta E_{\mathrm{CCD}}^{(i)}-\Delta E_{\mathrm{CCD}}^{(i-1)} \le \lambda \).
- It is common during the iterations to scale the amplitudes with a parameter \( \alpha \), with \( \alpha \in (0,1] \) as \( t^{(i)}=\alpha t^{(i)}+(1-\alpha)t^{(i-1)} \).