Vibrational Configuration Interaction Using a Tiered Multimode

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Vibrational Configuration Interaction Using a Tiered Multimode Scheme and Tests of Approximate Treatments of Vibrational Angular Momentum Coupling: A Case Study for Methane Steven L Mielke, Arindam Chakraborty, and Donald G. Truhlar J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/jp4011789 • Publication Date (Web): 08 Apr 2013 Downloaded from http://pubs.acs.org on April 13, 2013

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The Journal of Physical Chemistry

4/3/13

Vibrational Configuration Interaction Using a Tiered Multimode Scheme and Tests of Approximate Treatments of Vibrational Angular Momentum Coupling: A Case Study for Methane Steven L. Mielke,* Arindam Chakraborty,‡ and Donald G. Truhlar* a Department

of Chemistry and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, MN 55455-0431, USA

Abstract. We present vibrational configuration interaction calculations employing the Watson Hamiltonian and a multimode expansion. Results for the lowest 36 eigenvalues of the zero total angular momentum rovibrational spectrum of methane agree with the accurate benchmarks of Wang and Carrington to within a mean unsigned deviation of 0.68, 0.033, and 0.014 cm−1 for 4-mode, 5-mode, and 6-mode representations, respectively. We note that in the case of the 5-mode results this is a factor of 10 better agreement than for 5-mode calculations reported earlier by Wu, Huang, Carter, and Bowman, for the same set of eigenvalues, indicating that the multimode expansion is even more rapidly convergent than previously demonstrated. Our largest calculations employ a tiered approach with matrix elements treated using a variable order multimode expansion with orders ranging from 4-mode to 7-mode; strategies for assigning matrix elements to particular multimode tiers are discussed. Improvements of 7-mode coupling over 6-mode coupling are small (averaging 0.002 cm−1 for the first 36 eigenvalues) suggesting that 7-mode coupling is sufficient to fully converge the results. A number of approximate treatments of the computationally expensive vibrational angular momentum terms are explored. The use of optimized vibrational quadratures allows rapid integration of the matrix elements, especially the vibrational angular momentum terms, which

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2 require significantly fewer quadrature points than are required to integrate the potential. We assign the lowest 243 states and compare our results to those of Wang and Carrington, who provided assignments for the same set of states. Excellent agreement is observed for most states, but our results are lower for some of the higher-energy states by as much as 20 cm−1, with the largest deviations being for the states with six quanta of excitation in the F2 bends, suggesting that the earlier results were not fully converged with respect to the basis set. We also provide corrections to several of the state assignments published previously.

Keywords: Watson Hamiltonian, rovibrational spectrum, optimized vibrational quadrature, multimode expansion

*email: [email protected], [email protected] ‡Present

address: Department of Chemistry, Syracuse University, NY 13244


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3 1. Introduction Many ideas having roots in electronic structure theory have also been employed in solving for the vibrational spectrum of polyatomic molecules. Early examples of these include the vibrational self-consistent field (VSCF)1-13 and vibrational configuration interaction (VCI) methods,1,7,8,12-31 and a large volume of more-recent work32-45 has drawn upon a wide array of such ideas. One particularly important recent advance has been the introduction of the multimode representation scheme,25,46 which allows higherdimensional systems to be treated with manageable cost and high accuracy. This representation scheme was employed in the MULTIMODE code of Carter, Bowman, Handy and coworkers,12,47 which has been used extensively.12,13,29,34,47-55 In earlier work by two of the present authors30,31 the Watson and vibrational angular momentum (VAM) terms in the Watson Hamiltonian56 were neglected, which is a common approximation, and multimode (MM) expansions were limited to 4-mode treatments. We have now implemented these terms in the latest version of our code, which was also extended to consider MM expansions through seventh order. A straightforward implementation of the VAM terms can greatly increase the computational cost so we have developed a number of strategies to make their inclusion affordable and we have also explored a number of different approximation strategies involving these terms. In order to test and illustrate these methods we have performed calculations for methane because highly accurate benchmarks are available for this system.54,57-59



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4 2. Background Our calculations use basis functions that are products of one-dimensional harmonic oscillator (HO) functions obtained from a normal mode analysis at the equilibrium configuration. Some VCI methods12,13,60 take the basis functions as products of the individual-mode wave functions, called modals, that are obtained by VSCF calculations obtained using the full Hamiltonian, and such modals do lead to improved accuracy for the lowest eigenvalue, but when one is interested in converging many rovibrational states there is little reason to believe that the virtual modals of a groundstate VSCF calculation form an especially appropriate basis. Working directly with the primitive basis functions leads to simpler code and allows for simple schemes to identify matrix elements of less significance that can be treated with lower-order multimode expansions. Our calculation of the Hamiltonian matrix is parallelized by distributing the evaluation of the matrix elements among the available processors. However, the work involved in calculating various matrix elements varies markedly, so assigning an equal number of elements to each processor would lead to very poor load balance. Instead, we estimate the work involved in calculating each of the matrix elements and then assign elements so that each processor is tasked with an approximately equal amount of work. Except for low orders of the MM expansion, nearly all of the computational effort is expended in forming the Hamiltonian matrix. For convenience, in the present set of calculations the matrix diagonalization of the calculated Hamiltonian matrix is performed in a separate calculation employing a single computer node. For larger calculations parallel diagonalization or even the use of iterative eigensolvers61 may be necessary.


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5 In the following we review the theoretical background on which the calculations are based and the numerical methods used to make them affordable.

2.1. The Watson Hamiltonian. The Watson Hamiltonian56 is given by !2 !2 1 3 " + " (J" ! #" )µ"# (J # ! # # ) 2 k ! Qk2 2 " , # =1 + U(Q1,Q2 ,…QN ) + V (Q1,Q2 ,…QN )

H =!

(1)

where

!! = "" st(! )Qs Pt

(2)

st

The Qi are normal coordinates defined to be zero at the equilibrium configuration, N is the number of vibrational normal mode coordinates, Pi is the momentum conjugate to coordinate Qi, i.e.,

Pi = !i!

" "Qi

(3)

J! and J ! are components of a vector operator J representing the total angular momentum referred to the moving axes. U is the so-called Watson term, given by

1 U(Q1,Q2 ,…QN ) = ! ! 2 " µ!! 8 !

(4)

where µ is the inverse effective moment of inertia matrix, with elements given by

( )!"

µ!" = I ! "1

where the effective moment of inertia matrix I! has elements given by


(5)


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6 (! ) ( " ) I!" # ts QrQt ! = I!" " ## rs

(6)

rst

and where I is the ordinary moment of inertia matrix. The Coriolis coupling constants are given by

! st(" ) = !! ts(" ) = " #"$% " l$ i,s l% i,t

%$(7)

i

where l! i,s is a component of the normal mode eigenvector for mode s, and !"#$ is the Levi-Civita symbol. For the special case of J = 0, eq 1 reduces to

H =!

3 !2 !2 1 + " " " " rs(# )" tu($ )µ#$ Qr PsQt Pu 2 k ! Qk2 2 rstu # , $ =1

!2 3 ! " µ!! + V (Q1,Q2,…QN ) 8 ! =1

(8)

The term !! is conventionally56 referred to as the vibrational angular momentum, although it has long been understood62 that the term contains an additional rotational contribution. We will refer to the second and third terms in eq 8 as the vibrational angular momentum (VAM) and Watson terms, respectively.

2.2 The Multimode Expansion. The M-mode MM expansion represents the potential in the form

V (Q1,Q2 ,…,QN ; M ) = V

(0)

N

+ !V i N

+... +

!

(1)

N

(Qi ) + ! V (2) (Qi ,Q j ) i< j

V (M ) (Qi ,Q j ,...)

i< j

Vibrational Configuration Interaction Using a Tiered Multimode

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