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1żManzhos Sergei !"# $% 1!Carrington Tucker2&'()()*+),-./ 1!Queen's University, Kingston ON, Canada2
We address the problem of computing a vibrational spectrum using a small numberof basis functions in the expansion of the wavefunction and a small number of samples ofthe potential energy surface (PES). While it is possible today to solve for hundreds of vlevels of a 5-6 atomic molecule (in vacuum) with high accuracy, the task is formidable dueto the size of the Hamiltonial matrix. The most difficult calculations to date implicate
matrices with as many as hundreds of millions of rows. This also requires the knowledge ofthe PES at millions of locations, so a PES function is necessary, the construction of which isalso problematic.
In many applications, however, only several v-levels are desired, and the accuracy <0.1 cm-1 is not required. For example, in applications ranging from heterogeneous catalysis
to selective uptake of molecules by nanotubes, it is important to identify species andadsorbate complex geometries. This can be done using IR spectra, as they are very sensitive
to geometry and can be used to identify species and understand structure when it cannot be
assigned based on energy alone. Spectra can serve as proxy to catalytic activity. Spectra canalso be used for nanotube quality assessment (Figure 1). A complex with an adsorbedmolecule has a much higher dimensionality than the molecule in vacuum, which is why
today, only normal mode, harmonic analysis of frequencies is practical. However, the PESof adsorbate-containing systems deviates strongly from a (uncoupled) harmonic well. A
method is desired to compute spectral signatures of molecule-surface/particle complexes
taking into account the real shape of the PES.
Surf. Sci

Appl. Phys. Lett

Figure 1. Adsorbate complexes are one example where computing IR frequencies is important forspecies identification. However, only DFT’s harmonic analysis is practical today.

We present a method that calculates several vibrational levels from a very small,
flexible basis of neurons of a neural network (NN). It simplifies the solution of theSchrödinger equation in many dimensions, where basis size explosion (due mainly to theuse of direct product bases) is the principal impediment to studying larger molecular andreactive systems. The wavefunction is represented in the form of a radial-basis NN

1Nζ§b·d§d·2
ψ k ( y ) = ¦¦ cisk ¨ isk ¸ exp ¨ −bisk( wij − y j ) ¸¦=1
i =1 s =1j© π ¹©
¹2(1)It is an expansion over N parameterized (flexible) basis functions each consisting ofζ Gaussian terms with the same center but different and adjustable widths. This allows awide range of shapes of the basis functions with few additional parameters, see Figure 2.

Figure 2. Various shapes of the basis functions are possible with multi-ζ Gaussian basis.

Expansion (1) automatically satisfies boundary conditions. In contrast to previous
attempts at NN-eigensolvers, our approach combines non-linear optimization of parametersof the basis of neurons (blue in Eq. (1)) with a method to solve a rectangular matrix problem
for linear parameters (red in Eq. (1)), which improves dimensionality scaling and allows forthe output of many levels at the same time. Requiring that Eq. (1) satisfy the Schrödinger
equation at a set of M points in the configuration space leads to a rectangular matrixproblem(MM×ȗN− Ek S M×ȗN ) ck = 0(2)where S is the overlap matrix and M depends on the Hamiltonian (problem-dependent). Eq.
(2) is solved for the linear parameters c while the non-linear basis parameters b and w arefitted.
The algorithm [Can. J. Chem. 87, 864 (2009), Chem. Phys. Lett., 474, 217 (2009)]
avoids the calculation of integrals and of the potential energy function. We demonstrate its
application to a model problem in up to 6 dimensions and to the spectrum of water. Only a
few dozens of neurons are needed to compute five levels of water from a small set ofsamples of the potential surface (Figure 3).

Figure 3. Example: 2D cuts of the third vibrational level of H2O. In this calculation, 5 v-levels wereobtained from only 15 flexible (ζ=3) basis functions and 3000 samples of the potential
surface.

2