wiki:VMFCI

The VMFCI Method explained

VMFCI is the acronym for Vibrational Mean Field Configuration Interactions. The method consists in performing vibrational configuration interactions of some degrees of freedom (dof) in the field of the other. The aim of the method is to keep the size of the finite basis sets used in successive VCI manageable by contracting groups of dof together, as in the traditional contraction method [1]. However, the power of the method comes from the mean field term added to the group Hamiltonians not present in previous approaches.

A VMFCI step starts with a partition, $P$, of the nvib vibrational degrees of freedom into np subsets :

$$P=(I_1,I_2,\cdots,I_{n_P})=(\{i_{1}^{1},i_{2}^{1},\cdots,i_{k_1}^{1}\},\{i_{1}^{2},i_{2}^{2},\cdots,i_{k_2}^{2}\},\cdots,\{i_{1}^{n_P},i_{2}^{n_P},\cdots,i_{k_{n_P}}^{n_P}\})\quad (1) $$

Using partition $P$ the vibrational Hamiltonian can be written as :

$$H_{vib}  = h_0 + \sum_{\gamma_1=1}^{n_P} h_{\gamma_1}(I_{\gamma_1}) + \sum_{1\leq \gamma_{1}<\gamma_{2}\leq n_P} h_{\gamma_{1},\gamma_{2}}(I_{\gamma_{1}})h_{\gamma_{1},\gamma_{2}}(I_{\gamma_{2}})+\cdots + h_{1,2,\cdots ,n_P}(I_{1})h_{1,2,\cdots ,n_P}(I_{2})\cdots h_{1,2,\cdots ,n_P}(I_{n_P}) $$

where

$h_{\gamma_1,\gamma_2,\ldots ,\gamma_k}(I_{\gamma_l})$

denotes a vibrational operator which only depends upon operators acting on the degrees in subsets I$_{\gamma_l}$.

Then, one defines a possibly coarser partition, $Q$ :

$$Q=(J_1,J_2,\cdots,J_{n_Q})\quad  satisfying \quad n_Q\leq n_P \quad and \quad \forall  \gamma\in\{1,\cdots ,n_P\},  \quad  \exists \alpha\in\{1,\cdots ,n_Q\}\quad such\ that\quad I_\gamma\subseteq J_{\alpha}$$

For such a step, we call contractions the subsets J$\alpha$ and components of contraction J$\alpha$, the subsets I$_\gamma$ such that :

$$I_\gamma\subseteq J_\alpha$$

Notes

  1. The components of one step are the contractions of the previous step.
  1. $Q=P$ is allowed: Iterating the same partition until self-consistency lead to VSCFCI methods which generalizes

the well-known VSCF method [2,3]. Such a generalization has been considered by Bowman and Gazdi [4] but not in the frame of an iterated VCI approach.

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Let us consider one given contraction J$\alpha$ . We assume that contraction J$\alpha$ has $\beta$ components :

$$J_{\alpha}=I_{\gamma_1}\cup I_{\gamma_2}\cup\cdots\cup I_{\gamma_{\beta}}=\left\lbrace i_{1}^{\gamma_1},\cdots,i_{k_{\gamma_1}}^{\gamma_1},\cdots,i_{1}^{\gamma_{\beta}},\cdots,i_{k_{\gamma_{\beta}}}^{\gamma_{\beta}}\right\rbrace=\left\lbrace j_1^{\alpha},\cdots,j_{l_\alpha}^\alpha\right\rbrace \quad \text{with} \quad l_\alpha=k_{\gamma_1}+\cdots+k_{\gamma_{\beta}}.$$

Having a basis set spanning an Hilbert subspace of dimension, say d$_\gamma$, for each component I$_\gamma$:

$$\{\phi^{m_\gamma}_{I_\gamma}\}_{{m_\gamma}\in \{{1,\ldots ,d_\gamma}\}}, $$

we build for contraction J$\alpha$, a so-called product basis set, spanning an Hilbert subspace of dimension D$\alpha$,

$${\{\Phi^{M_{\alpha}}_{J_{\alpha}}\}}_{M_{\alpha}\in \{{1,\ldots ,D_{\alpha}}\}}$$

by constructing product functions of the form:

$$\Phi^{M_{\alpha}}_{J_{\alpha}}=\bigotimes\limits_{I_\gamma\subseteq J_{\alpha}}\phi^{m_\gamma}_{I_\gamma}\quad (2)$$

or more explicitly using variables dependencies:

$$\Phi^{M_{\alpha}}_{J_{\alpha}}(q_{i_{1}^{\gamma_1}},\cdots,q_{i_{k_{\gamma_1}}^{\gamma_1}},\cdots,q_{i_{1}^{\gamma_{\beta}}},\cdots,q_{i_{k_{\gamma_{\beta}}}^{\gamma_{\beta}}})=\Phi^{M_{\alpha}}_{J_{\alpha}}(q_{j_{1}^\alpha},\cdots,q_{j_{l_\alpha}^\alpha})=\prod\limits_{I_\gamma\subseteq J_{\alpha}}\phi^{m_\gamma}_{I_\gamma}(q_{i_{1}^\gamma},\cdots,q_{i_{k_\gamma}^\gamma})\quad with \quad M_{\alpha}=(m_1,\cdots ,m_{\beta})$$

Notes

  1. In CONVIV, the process is initialized with an initial basis set of modals, that is to say, functions of a single vibrational degree of freedom, but this constraint can be walked around if groups of dof have to be contracted from the start. The modals can be eigenfunctions of one dimensional Schrödinger equations with harmonic potential (with arbitrary center and frequency), Morse potential, Trigonometric Pösch-Teller potential, Kratzer potential, or can be Chebychev polynomials.
  1. By convention, the eigenstates are numbered in increasing order with natural numbers starting with 0, so the vector index M$\alpha$=(0, ... ,0) always corresponds to product of ground state functions.
  1. The dimension of the basis set for the contraction J$\alpha$ can be different from the product of dimensions of its component's basis sets because of possible basis function truncations, usually performed according to some energy criteria (see infra).

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For the contraction J$\alpha$, we define a partial Hamiltonian, H$\alpha$, by grouping all the terms in Hvib involving the degrees in components Ijl ofJ$\alpha$:

$$H_\alpha  = h_0 + \sum\limits_{\stackrel{\gamma_1}{\stackrel{such\ that}{I_{\gamma_1}\subseteq J_\alpha}}}h_{\gamma_1}(I_{\gamma_1})+ \sum\limits_{\stackrel{\gamma_{1}<\gamma_{2}}{\stackrel{such\ that}{I_{\gamma_1},I_{\gamma_2}\subseteq J_\alpha}}}h_{\gamma_{1},\gamma_{2}}(I_{\gamma_{1}})h_{\gamma_{1},\gamma_{2}}(I_{\gamma_{2}}) +\cdots+\sum\limits_{\stackrel{\gamma_{1}<\cdots <\gamma_{\beta}}{\stackrel{such\ that}{I_{\gamma_1},\cdots ,I_{\gamma_{\beta}}\subseteq J_\alpha}}}h_{\gamma_{1},\cdots ,\gamma_{\beta}}(I_{\gamma_{1}})\cdots h_{\gamma_{1},\cdots ,\gamma_{\beta}}(I_{\gamma_{\beta}})$$

In contrast with the original contraction method, a term accounting for the mean field of all the modes not in contraction J$\alpha$, called spectator modes, is added to this partial Hamiltonian:

$$\widetilde{H_\alpha}=H_\alpha + \langle \bigotimes\limits_{I_\gamma\nsubseteq J_\alpha}\phi^{0}_{I_\gamma}|H_{vib} - H_\alpha | \bigotimes_{I_\gamma\nsubseteq J_\alpha}\phi^{0}_{I_\gamma}\rangle \quad (3)$$

The VMFCI step consists in performing a vibrational configuration interaction [5] (Galerkin method) for each mean field Hamiltonians, Eq.(3), in the product basis sets, Eq.(2).

Thereby, we obtain new basis sets of dimension D$_\alpha$ made of eigenvectors of the mean field Hamiltonians:

$\quad \{\phi^{m_\alpha}_{J_\alpha}\}_{m_\alpha\in\{1,\ldots, D_\alpha\}}$,

to construct the product basis sets of the next VMFCI step. Their associated eigenvalues can be used to truncate the product basis sets according to energy criteria: either the individual component basis functions are selected only if their associated eigenvalue is less than a given threshold, or a product function is selected only if the sum of its component basis function eigenvalues is less than a given threshold. Both criteria can be applied together.

Notes

  1. In CONVIV, it is possible to control the number of eigenpairs calculated by the diagonalizer when

solving the mean field Hamiltonian eigenvalue problem. So, in fact, the dimension of the new basis set for contraction J$\alpha$, can be less than D$\alpha$.

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Last modified 8 years ago Last modified on Sep 13, 2011 2:29:54 PM