A high performance grid-based algorithm for computing QTAIM properties

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Abstract

An improved version of our method for computing QTAIM [J.I. Rodríguez, A.M. Köster, P.W. Ayers, A. Santos-Valle, A. Vela, G. Merino, J. Comput. Chem. (2009), in press, doi:10.1002/jcc.21134] is presented. Vectorization and parallelization of the previous algorithm, together with molecular symmetry, make the present algorithm as much as two orders of magnitude faster than our original method. The present method scales linearly with both system size and the number of processors. The performance of the method is demonstrated by computing the QTAIM atomic properties of a series of carbon nanotubes. Our results show that the CPU time for a QTAIM property calculation is comparable to that of a SCF-single point calculation. The accuracy of the original method is also improved.

Graphical abstract

The CPU time for a QTAIM property calculation can be reduced to that of a SCF-single point calculation.

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Introduction

Atomic properties are useful for interpreting the results of electronic structure calculations on molecules and materials. (For example, atomic charges and multipole moments provide valuable information about intermolecular forces.) An appropriate definition of the properties of atoms inside molecules, based on quantum mechanical principles, is provided by the quantum theory of atoms in molecules (QTAIM), the physics of an open system [1].

The importance of QTAIM properties has been widely recognized in the quantum chemistry and solid state physics communities [2], [3], [4]. QTAIM has been extensively exploited in fields ranging from solid state physics and X-ray crystallography to drug design and biochemistry [4]. However, it is not always feasible to apply QTAIM to large systems due to its computational cost. (QTAIM property calculations are often several orders of magnitude slower than a SCF-single point calculation.) Researchers developing more efficient QTAIM computer algorithms have explored a variety of intricate numerical schemes [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19]. The basic concepts of grid-based algorithms date back to the early days of QTAIM [13], [14], but these methods have received renewed attention due to their easy numerical implementation and relatively high efficiency [15], [16], [17], [18], [19]. Although recent grid-based methods seem less suitable for high accuracy calculations than the standard approaches, they can be about 30 times faster than the standard approaches [18], [19]. Our goal is to reduce the computational cost of a QTAIM calculation so that it is comparable to an SCF-single point calculation.

In this Letter, we present an improved version of the grid-based method we recently introduced [19]. This improved algorithm was implemented in the Amsterdam Density Functional program (ADF) [20], [21], [22]. There are four main improvements in the new implementation: (I) Accuracy of the integration method. (II) Vectorization of the procedure for constructing the steepest ascent paths of the density. (III) The use of the molecular symmetry. (IV) Parallelization. The algorithm is presented in Section 2 (overview of the previous algorithm) and Section 3 (new developments). Sections 4 Computational methods and hardware, 5 Results and discussion contain the details about our computational tests.

Section snippets

The original grid-based method

Within QTAIM, a property PA of the atom A is defined as the expectation value of an effective single-particle property density p(r) over its so-called atomic basin ΩA[2], [3], [4],PA=ΩAp(r)dr.The atomic basins must be determined and may have complicated shapes, which makes the integration in Eq. (1) difficult. The explicit numerical construction of the boundaries of the atomic basins, the zero-flux surfaces, is the bottleneck in most QTAIM software [5], [6], [7], [8], [9], [10], [11], [12].

Accuracy

We introduced the virial factor [2], [26] for accurately computing the QTAIM energies (see Table 1). The accuracy in computing the other atomic properties (charges, dipole and quadrupole moments) was also improved by using the ADF symmetry adapted grids [20], [21], [22], [27] (see Table 1).

Vectorization

In ADF the grid points are divided in NB blocks (sets), each with NBP grid points. This is done to facilitate both automatic vectorization in inner loops over the points in a block and the parallelization of

Computational methods and hardware

All calculations were performed with a development version of ADF using a TZP Slater basis. We used the Dirac exchange functional [24] with the Vosko–Wilk–Nusair correlation functional [25]. Unless otherwise stated, the ADF default settings for the SCF procedure were used [20], [21], [22]. The 2nd order Runge–Kutta algorithm with a step size equal to 0.2 a.u. was used for constructing the steepest ascent paths of the density [19], [26]. The initial radius of each atomic trust sphere as defined

Results and discussion

A single point calculation at the optimized geometry3 was used to generate the electron density for our atomic property calculations. Table 1 shows the QTAIM atomic properties for some representative molecules. From Table 1, we can see that the accuracy of our

Conclusions

We report, for the first time, an algorithm for computing the properties of atoms (as defined through the quantum theory of atoms in molecules) that has computational cost comparable to a standard single point LDA/GGA Kohn–Sham DFT calculation. This breakthrough is built upon our previous grid-based method for computing atomic properties (thus avoiding the explicit construction of the zero-flux surfaces), but the implementation presented here is much faster. The increase in speed is due to

Acknowledgements

We would like to thank Dr. Erik Van Lenthe, Dr. Pier Philipsen, Dr. Alexei Yakovlev, Dr. Stan Van Gisbergen, and Dr. Todd Keith for useful discussions about this work and for proofreading this manuscript. J.I.R. gratefully acknowledge the Vrije Universiteit Amsterdam theoretical chemistry group and all SCM people for hosting him during the development of this work and for sharing their computer facilities. J.I.R. also thanks Miss. Silvia Dimitrova for her insightful discussions and for LMBABB.

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