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A linear- and sublinear-scaling method for calculating NMR shieldings in atomic orbital-based second-order Møller-Plesset perturbation theory

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10.1063/1.4801084

### Abstract

An atomic-orbital (AO) based formulation for calculating nuclear magnetic resonance chemical shieldings at the second-order Møller-Plesset perturbation theory level is introduced, which provides a basis for reducing the scaling of the computational effort with the molecular size from the fifth power to linear and for a specific nucleus to sublinear. The latter sublinear scaling in the rate-determining steps becomes possible by avoiding global perturbations with respect to the magnetic field and by solving for quantities that involve the local nuclear magnetic spin perturbation instead. For avoiding the calculation of the second-order perturbed density matrix, we extend our AO-based reformulation of the Z-vector method within a density matrix-based scheme. Our pilot implementation illustrates the fast convergence with respect to the required number of Laplace points and the asymptotic scaling behavior in the rate-determining steps.

© 2013 AIP Publishing LLC

Received 23 January 2013
Accepted 26 March 2013
Published online 01 May 2013

Acknowledgments: The authors thank Dr. M. Beer, Dr. J. Kussmann, and S. A. Maurer (University of Munich, LMU) for useful discussions. C.O. acknowledges financial support by the Volkswagen Stiftung within the funding initiative “New Conceptual Approaches to Modeling and Simulation of Complex Systems,” by the SFB 749 “Dynamik und Intermediate molekularer Transformationen” (DFG), and the DFG cluster of excellence (EXC 114) “Center for Integrative Protein Science Munich” (CIPSM).

Article outline:

I. INTRODUCTION

II. THEORY

A. AO-MP2 energy expression

B. Laplace-based AO-MP2 chemical shielding tensor

1. The first derivative with respect to the magnetic field**B**

2. Second derivatives by further differentiation: The derivative with respect to the nuclear magnetic spin moment **m** _{ j }

3. AO-based Z-vector method for avoiding the second-order density matrix

C. Nuclei-selected AO-MP2 shieldings—A sublinear scaling reformulation

III. FIRST PRELIMINARY IMPLEMENTATION AND SCALING BEHAVIOR

A. Required number of Laplace points

B. Locality of MP2 shieldings and comparison to HF shieldings

C. Expected scaling behavior for calculating the AO-MP2 shielding tensor

1. Scaling behavior of transformations with perturbed pseudo-densities

2. Scaling behavior of the unperturbed R-matrices

3. Scaling behavior of the Z-vector equations for the first and second derivative

IV. CONCLUSION

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2013-05-01

2014-04-19

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