Chapter 11
Vibrational Excited States by Diffusion Monte Carlo
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Anne B. McCoy Department of Chemistry, The Ohio State University, Columbus, O H 43210
Methods for evaluating vibrationally excited states by D M C within the fixed-node approximation are described. Two central issues are identified. The first, and most important, is the identification of the coordinates that will be used to describe the molecular vibrations. The second is the determination of where the node should be placed. Strategies for addressing both of these issues are discussed within the context of the fundamental vibrations in N e O H and H O . -
2
3
2
Introduction One of the greatest challenges for and limitations of the Diffusion Monte Carlo ( D M C ) approach is the treatment of nodes. This is clearly a problem for any application of D M C to electronic structure problems. For vibrational problems, even i f the ground state is nodeless, it is unusual for that to be the only state of interest, and excited states all contain nodes. In cases where the functional form of the ΛΜ dimensional nodal surface is known, the solution is straightforward, since the behavior of a wave function in the vicinity of a node is identical to that when the potential is forced to be infinite along this surface (/). Specifically, the wave function goes to zero at the node, and its first derivative is finite and constant across the node. This can be readily seen in one-dimension. Consider the first excited state for the harmonic oscillator potential, V (r) = kl2 r . The energy of this state is the same as the energy of the ground state wave function of potentials that are 1
x
© 2007 American Chemical Society
In Advances in Quantum Monte Carlo; Anderson, J., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2006.
147
148 defined as
00
y*)
r 0
(1)
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and
r0 In addition, the wave function that corresponds to the first excited state solution to the Schrodinger equation, where the full harmonic oscillator potential is used, is identical, within a constant factor, to the ground state solution for the potential in Eq. (1) for r > 0 and the ground state solution for the potential in Eq. (2) for r
