FanPT (Perturbation Theory for Parameterized Projected Schrodinger Equation)

[PaulWAyers]

We are implementing FanPT. The basic idea is in the recent paper.

Recall that the FanCI equations are

$$ \braket{\mathbf{m}|\hat{H}|\Psi(\mathbf{c})} = E \braket{\mathbf{m}|\Psi(\mathbf{c})} \qquad \qquad \forall \mathbf{m} \in \mathcal{P} $$

where $\mathcal{P}$ is the projection space (user-specified), $\mathbf{c}$ are the wavefunction parameters, and the nonlinear FanCI equations are usually overdetermined and solved in a least-squares sense. The set of determinants $\mathbf{n} \in \mathcal{S}$ is defined as the space of determinants that are linked to determinants in $\mathcal{P}$ by the Hamiltonian, typically just single- and double-replacements of the occupied orbitals.

We write the Hamiltonian as a zeroth-order part (for which the FANCI equations have been solved) and a first-order part,

$$ \hat{H}(\lambda) = \hat{H}(\lambda_0) + (\lambda - \lambda_0) \hat{H}'(\lambda_0) + \tfrac{1}{2} (\lambda - \lambda_0)^2 \hat{H}''(\lambda_0) +\cdots $$

Going through some math, we get the equation

$$ \sum_{\mathbf{n} \in \mathcal{S}} f_\mathbf{n}(\lambda_0,\mathbf{c}) E'(\lambda_0) + \sum_{\mathbf{n} \in \mathcal{S}} \left(\braket{\mathbf{m}|\hat{H}(\lambda_0)|\mathbf{n}} - E(\lambda_0) \delta_{\mathbf{m} \mathbf{n}}\right) \frac{d f_{\mathbf{n}}(\mathbf{c})}{d \mathbf{c}(\lambda_0)} \cdot \mathbf{c}'(\lambda_0) = \sum_{\mathbf{n} \in \mathcal{S}} \braket{\mathbf{m}|\hat{H}'(\lambda_0) |\mathbf{n}} f_\mathbf{n}(\lambda_0,\mathbf{c}) $$

The key ingredients are the Hamiltonian matrix elements. The right hand side is easy to evaluate in terms of transition-2DMs, which appear on the right-hand-side,

$$ \text{RHS} = \text{Tr}[\hat{H}'(\lambda_{0})_{pqrs} \Gamma_{pqrs}(\lambda_{0})] $$

My tendency would be to implement this as some functions that build the linear equations to solve, and then solve them. I realize that the (old) implementation is somewhat more general, but we may also want to focus on a specific one or two flavors if it makes the code a lot simpler/faster. Ramon, what do you think the best flavors are? (It will depend on how much computational complexity and code complexity we add by writing a general treatment.)

[carlosevmoura]

Initial implementation of FanPT in PyCI #70

This initial implementation is based on the original FanPT code written by Ramon's Group to the pilot code Fanpy. Here is a brief description of the code:

pyci/fanci/fanpt_wrapper.py

Contains the FanPT solver (solve_fanpt) function and additional functions.

pyci/fanpt/base_fanpt_container.py

Defines general FanPT class (FANPTContainer). It is used by derived classes FANPTContainerEParam and FANPTContainerEFree.

pyci/fanpt/fanpt_cont_e_param.py

Define a FANPTContainerEParam class, which contains the data and methods required for a FanPT calculation using energy as a parameter.

pyci/fanpt/fanpt_cont_e_free.py

Define a FANPTContainerEFree class containing the data and methods required for an energy-free FanPT calculation. It is derived from FANPTContainerEParam.

pyci/fanpt/fanpt_constant_terms.py

Define the FANPTConstantTerms class, which contains the constant terms of the FanPY system of equations and methods.

pyci/fanpt/fanpt_updater.py

Contains the FANPTUpdater class, which is used by the FanPT solver to update FanCI parameters and energy.

pyci/test/test_fanpt.py

Includes two tests for FanPT solver, using Energy Free and Energy as Parameter flavors.

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There's room for improvement in code design and file organization, especially in matching the PyCI interface for seamless integration.
Let me know if you have suggestions, and I will work on them.

[PaulWAyers]

@msricher and @marco-2023 can you look at this and see what the next step should be? I have a feeling some of the data classes may be better off as named tuples.

[msricher]

I've looked at it and I think this is fine the way it is. The next step might be making sure that pCCD+S can now be reliably solved, so that I can do a better test of the stochastic optimization?

[PaulWAyers]

Ok, sounds good. Thanks for looking at it @msricher .

I agree with your next step.

@carlosevmoura , we'll do a formal code review on it (I haven't seen the pull request but I'm way behind). That will usually lead to some changes. But then we will have a functional version. I anticipate a refactor at some point of a lot of FanCI stuff, but I am too overwhelmed to think about that for now.