An adaptive density-guided approach for the generation of potential energy surfaces of polyatomic molecules

Abstract

We present an adaptive density-guided approach for the construction of Born–Oppenheimer potential energy surfaces (PES) in rectilinear normal coordinates for use in vibrational structure calculations. The procedure uses one-mode densities from vibrational structure calculations for a dynamic sampling of PESs. The implementation of the procedure is described and the accuracy and versatility of the method is tested for a selection of model potentials, water, difluoromethane and pyrimidine. The test calculations illustrate the advantage of local basis sets over harmonic oscillator basis sets in some important aspects of our procedure.

Appendix

1.1 Integrals for distributed Gaussian basis sets

Gaussian integrals can be easily done analytically from a few standard integrals ($\int_{-\infty}^\infty e^{-\lambda x^2}\,{\rm d}x=\sqrt{{\frac{\pi} {\lambda}}}, \int_{-\infty}^\infty x^{n} e^{-\lambda x^2}\,{\rm d}x=(-{\frac{\rm d} {{\rm d}\lambda}})^{n/2}\sqrt{{\frac{\pi} {\lambda}}}$ with n an even positive integer number) and the fact that the product of two Gaussians is another Gaussian:

$$ G_{ij}(x) = G_{i}(x)G_{j}(x) = \left({\frac{2\zeta_{ij}} {\pi}} \right)^{{\frac{1} {4}}} \exp(-\zeta_{ij}(Z_{i}-Z_{j})^2) \left({\frac{2(\zeta_{i} + \zeta_j)} {\pi}}\right)^{{\frac{1} {4}}} \exp(-(\zeta_{i}+\zeta_j) (x-Z_{ij})^2) $$

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with

$$ \zeta_{ij} = {\frac{\zeta_{i} \zeta_{j}} {\zeta_{i} + \zeta_{j}}} $$

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$$ Z_{ij} = {\frac{\zeta_{i} Z_{i} + \zeta_{j} Z_{j}} {\zeta_{i} + \zeta_{j}}} $$

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The overlap integrals are

$$ O_{ij} = \langle {G_{i}} |{G_{j}} \rangle = \int\limits_{-\infty}^\infty G_{i}^{*}(x) G_{j}(x)\,{\rm d}x = \left({\frac{4\zeta_{ij}} {(\zeta_{i}+\zeta_{j})}}\right)^{({\frac{1} {4}})} \exp(-\zeta_{ij}(Z_{i}-Z_{j})^2) $$

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Clearly, the basis set of distributed Gaussians is a non-orthogonal basis with overlaps decaying fast with the distance between the centers of the Gaussians and with increasing exponents. Other integrals are

$$ (x^n)_{ij} = \langle {G_i} | {x^n} |{G_j} \rangle = \sum_{k=0}^{int(n/2)} {\left(\begin{array}{l}{n} \\ {2k} \end{array}\right)} {\frac{(2k-1)!!} {2^k(\zeta_{i}+\zeta_{j})^k}} Z_{ij}^{n-2k} O_{ij} $$

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$$ \left({\frac{\rm d} {{\rm d}x}}\right)_{ij}=\langle {G_i} | {{\rm d}/{\rm d}x} |{G_j} \rangle=2 \zeta_jO_{ij}(Z_j-Z_{ij}) $$

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$$ T_{ij} = -{\frac{1} {2}} \langle {G_i} | {\frac{d^2} {dx^2}} |{G_j} \rangle = \zeta_{j}O_{ij}\left[1-2\zeta_j\left({\frac{1} {2(\zeta_{i}+\zeta_{j})}}+Z_{ij}^2-2Z_{j}Z_{ij}+Z_{j}^2\right)\right] $$

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with $(2k-1)!! = (2k-1)\times(2k-3){\ldots}\times 1$ and $0!!=1.$

Other integrals have to be computed if the Hamiltonian contains Watson kinetic energy terms, which can be easily derived from the integrals listed above:

$$ \langle {G_i} | {x^n{\frac{{\rm d}} {{\rm d}x}}} |{G_j} \rangle = -2\zeta_{j} \langle {G_i} | {x^{n+1}} |{G_j} \rangle+2\zeta_{j}Z_{j} \langle {G_i} | {x^n} |{G_j} \rangle $$

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$$ \langle {G_i} | {{\frac{{\rm d}} {{\rm d}x}}x^n} |{G_j} \rangle = n \langle {G_i} | {x^{n-1}} |{G_j} \rangle+\langle {G_i} | {x^n{\frac{{\rm d}} {{\rm d}x}}} |{G_j} \rangle $$

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$$ \begin{aligned} \langle {G_i} | {{\frac{{\rm d}} {{\rm d}x}}{x^n}{\frac{{\rm d}} {{\rm d}x}}} |{G_j} \rangle &= (4\zeta_{j}^2Z_{j}^2-2\zeta_{j}(n+1)) \langle {G_i} | {x^n} |{G_j} \rangle\\ &\quad + 2n\zeta_{j}Z_{j} \langle {G_i} | {x^{n-1}} |{G_j} \rangle\\ &\quad + 4\zeta_j^2\langle {G_i} | {x^{n+2}} |{G_j} \rangle-8\zeta_{j}^2Z_{j}\langle {G_i} | {x^{n+1}} |{G_j} \rangle \end{aligned} $$

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When fitting with scaled coordinates all these integrals are multiplied by a common factor of $(\sqrt{\omega})^n$ when the scaled coordinate x enters with a power of n.