# Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation

^{*}

## Abstract

**:**

## 1. Introduction

**r**denotes coordinates spanning the configuration space, $V\left(\mathit{r}\right)$ is the potential energy surface (PES), $\widehat{T}\left(\mathit{r}\right)$ is the kinetic energy operator (KEO), and $\psi \left(\mathit{r}\right)$ is the wavefunction. The values of the PES can be computed ab initio or sampled from an analytic function (which itself could be fitted to ab initio data or to empirical data) [2,3,4,5,6,7,8]. The KEO has a simple form in space-fixed Cartesian coordinates

**r**

^{SF}:

**c**):

**H**the Hamiltonian and

**B**the overlap matrix. Already with four atoms (a six dimensional configuration space), it is not unusual to use hundreds of thousands of basis functions (N), and the necessity to compute the integrals often requires PES values at millions of locations [16]. This, practically, requires availability of the PES as a continuous $V\left(\mathit{r}\right)$ function. Such functions are not trivial to construct with high accuracy, even for a four-atomic system, and become difficult to construct for larger systems [3,16,17,18]. For isolated molecules, it is relatively easy to compute a sufficient amount of high-quality ab initio data to which $V\left(\mathit{r}\right)$ can be fitted. For vibrations at interfaces and in materials, the cost of ab initio calculations is much higher, and for the vast majority of molecule-surface or aggregate-state systems of practical importance, there are no, and there will never be, ab initio based PES functions, even though vibrational problems with a sufficiently small number of coupled degrees of freedom can be identified in such systems [12,19,20].

#### Problem Statement and the Aim of This Work

^{−1}accuracy has been achieved for the vibrational spectrum of formaldehyde (a six-dimensional SE), with about 40,000 basis functions and with more than 100,000 collocation points [30,33,34]. While this kind of calculation is, in principle, doable on a modern workstation with a couple hundred GB of RAM, it is somewhat costly.

**c**controls the width of the function, and the exponent parameter β the degree of locality. It is usually assumed that β must satisfy β > d, the dimensionality of the space. This requirement is; however, made to insure integrability (of ${\theta}_{i}\left(\mathit{r}\right){\theta}_{j}\left(\mathit{r}\right)$ and ${\theta}_{i}\left(\mathit{r}\right)\widehat{H}{\theta}_{j}\left(\mathit{r}\right)$), and may not be necessary with collocation. The generalized IMQ function was previously used by Rabitz’s group to solve bound-state Schrödinger equations for modelling one- and two-dimensional problems (Morse oscillator and 2D Henon-Heiles potential) with the square (N = M) collocation method [35]. Hu et al. [35] concluded, based on those model problems, that the IMQ basis functions are advantageous for highly anharmonic problems and highly excited states. They also noted a slower rate of growth of the associated condition numbers compared to Gaussians, which bodes well for building a more complete basis.

## 2. Methods

_{2}CO, was solved for the lowest 100 levels in six bond coordinates, including the CO bond length, the two CH bond lengths, the two HCO angles, and the dihedral angle between the two HCO planes. The six coordinates, in this order, form the vector

**r**=

**r**

^{int}. The KEO was applied in space-fixed Cartesian coordinates (Equation (2)) using Equation (7), with a five-point finite difference stencil with all $d{x}_{k}=1\times {10}^{-5}$. See Reference [33] for details. The values of $V\left(\mathit{r}\right)$ at the collocation points were sampled from the analytic PES of Reference [36]. The collocation points $\left\{{\mathit{r}}_{\mathit{j}}\right\}$ were chosen within specific ranges of the six coordinates, from a pseudo-random six-dimensional Sobol sequence [37], and accepted into the collocation point set if

_{max}= 17,000 cm

^{−1}and Δ = 500 cm

^{−1}. The coordinate ranges were

**r**

_{min}= (1.03, 0.84, 0.84, 83, 83, 105),

**r**

_{max}= (1.50, 1.69, 1.69, 162, 162, 255), where bond lengths were in Å and angles in degrees. The point selection was; therefore, similar to that used in References [30,34] and thus allowed for comparison with results obtained with the Gaussian basis in those works.

**S**

^{T}

**S**on the order of 10

^{10}.

## 3. Results

#### 3.1. Effect of the IMQ Exponent

#### 3.2. Effect of the Basis Size

^{−1}or better. A similar result is observed with 100 levels, where IMQ outperforms for N = 15,000 and N = 20,000, but underperforms the Gaussian basis for larger basis sizes. For spectroscopically accurate calculations, the Gaussian basis is clearly preferred for both low-lying and highly excited states. The IMQ basis needs about a third more basis functions than the Gaussian basis to achieve a similar accuracy in this regime.

#### 3.3. Effect of the Rectangularity of the Collocation Equation

^{−1}.

## 4. Conclusions

^{−1}on the lowest 50 or 100 levels, more than N = 30,000 functions are needed and M > N is needed; for these values of N, the accuracy obtained with the IMQ basis is somewhat lower than that with the Gaussian basis.

^{−1}.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The mean absolute error (MAE), in cm

^{−1}, over the lowest 50 and 100 vibrational levels of H

_{2}CO, computed with the inverse multiquadratic (IMQ50, IMQ100) basis with different β (beta) parameters. The horizontal lines at 2 and 3.7 cm

^{−1}are corresponding values obtained with a Gaussian basis (G50, G100, respectively), as in Reference [30]. N = 20,000 basis functions and M = 80,000 collocation points were used.

**Figure 2.**The mean absolute error (MAE), in cm

^{−1}, over the lowest 50 and 100 vibrational levels of H

_{2}CO, computed with a width-optimized inverse multiquadratic (IMQ50, IMQ100) basis with β = 7 and a width-optimized multiquadratic Gaussian (G50, G100) basis, for different numbers of basis functions N and M = 3N collocation points. The insert shows part of the graph at the logarithmic scale.

**Figure 3.**The mean absolute error (MAE), in cm

^{−1}, over the lowest 50 and 100 vibrational levels of H

_{2}CO, computed with a width-optimized inverse multiquadratic (IMQ50, IMQ100) basis with β = 7 and a width-optimized multiquadratic Gaussian (G50, G100) basis, for N = 30,000 basis functions and different ratios of the number of collocation points to the number of basis functions M:N.

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Kamath, A.; Manzhos, S. Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation. *Mathematics* **2018**, *6*, 253.
https://doi.org/10.3390/math6110253

**AMA Style**

Kamath A, Manzhos S. Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation. *Mathematics*. 2018; 6(11):253.
https://doi.org/10.3390/math6110253

**Chicago/Turabian Style**

Kamath, Aditya, and Sergei Manzhos. 2018. "Inverse Multiquadratic Functions as the Basis for the Rectangular Collocation Method to Solve the Vibrational Schrödinger Equation" *Mathematics* 6, no. 11: 253.
https://doi.org/10.3390/math6110253