A novel extremely localized molecular orbitals based technique for the one-electron density matrix computation

doi:10.1016/j.cplett.2005.09.011

Chemical Physics Letters

Volume 415, Issues 4–6, 11 November 2005, Pages 256-260

https://doi.org/10.1016/j.cplett.2005.09.011 Get rights and content

Abstract

The ‘nearsightedness’ of electronic structure is an underlying principle in many of the linear scaling methods recently developed to study large systems. Among them, there are strategies based on the transfer of orbitals strictly localized on molecular fragments, such as the extremely localized molecular orbitals (ELMOs). Unfortunately, due to the non-orthogonal nature of these orbitals, the density matrix calculation is computationally demanding, so preventing a straightforward application to very large molecules. In this Letter, we show how this problem can be overcome by a proper application of the ‘Divide and Conquer’ strategy to the ELMO approach.

Introduction

The ‘nearsightedness’ of electronic structure [1], [2], that is the weak interaction between very distant regions of a large system, plays a crucial role in theoretical chemistry. Actually, exploiting this principle, several researchers have devised many linear scaling methods [3].

Among them, a possible strategy is the ‘Divide and Conquer’ (D & C) technique, initially developed by Yang [4] for the electron density and afterwards extended to the density matrix within the Hartree–Fock and the semiempirical MO approaches [5], [6], [7]. In these cases, after subdividing a large molecule into a set of relatively small subsystems with overlapping buffers, a proper electronic problem is solved for each region and then the total electron density (or density matrix) is assembled. A similar method, the molecular tailoring approach devised independently by Gadre et al. [8], [9], [10], consists in three steps: (1) fragments determination from the parent molecule; (2) ab initio density matrix calculation at the Hartree–Fock (or MP2) level for every subunit and (3) construction of the overall density matrix using the constituent fragment density matrices. In this context, the adjustable density matrix assembler (ADMA) strategy proposed by Exner and Mezey [11], [12], [13] is also noteworthy. This technique, which can be considered as an extension of the MEDLA approach [14], allows to obtain one-electron properties for macromolecules combining wisely the subsystem density matrices saved in a database previously determined by computations on smaller systems.

Another way of exploiting the nearsightedness of electronic structure to scale the computational cost is based on physically motivated reductions of the variational space, as the the ones recently performed by Janesco and Yaron [15] in the construction of their functional group basis sets.

Finally, it is important to consider the approach based on the transfer of localized molecular orbitals (LMOs) [16] which can be obtained by means of a posteriori [17] or a priori [18], [19], [20], [21], [22] techniques. The former consist in unitary transformations of canonical Hartree–Fock orbitals and preserve ‘orthogonalization tails’ that have to be deleted in order to make the resulting orbitals transferable [23]. The latter need the definition of a localization scheme before the calculation so that the molecular orbitals are expanded on the only basis functions centred on the atoms belonging to preselected fragments. In this way, the presence of tails is avoided a priori and orbitals strictly localized on few atoms, namely extremely localized molecular orbitals (ELMOs), are obtained.

In our laboratory, we have developed an algorithm to determine ELMOs [24] mainly based on the Stoll theory [18]. Furthermore, by means of extensive studies [20], [24], [25], [26], [27], [28], we have shown their reliable transferability that especially allows to reproduce very well one-electron properties at the Hartree–Fock level (e.g., electron density and molecular electrostatic potential).

It is worthwhile to note that both ELMOs and LMOs with deleted tails (LMOs-TD) are non-orthogonal orbitals. Hence, the calculation of the corresponding density matrix is not trivial and, for large systems, is very expensive in terms of CPU time. In order to reduce the computational cost, it is possible either to consider the orbitals as they were orthogonal or to orthogonalize them. However, in the first case a loss of accuracy naturally occurs, while the orthogonalization procedure becomes more and more demanding than using directly the non-orthogonal orbitals as the molecular size grows up.

So, in this Letter, we will propose a new strategy for an efficient density matrix computation for large systems. In particular, we will take advantage of the extremely localized nature of the ELMOs combined with the D & C philosophy to obtain an approximate density matrix expression that permits a considerable saving of the computational resources retaining the accuracy of the results.

Section snippets

Theory

Let us consider a 2N electrons closed-shell system described by a single Slater determinant built up with a set of N transferred ELMOs ${ϕ_{i}}_{i = 1}^{N}$ . The generic ith ELMO is a linear combination of a preselected subset of all the atomic orbitals, namely, $ϕ_{i} (r) = \sum_{μ = 1}^{NAO} C_{μ i} χ_{μ} (r),$ where according to the adopted localization scheme, some C_μi coefficients are constrained to be zero, while others have been variationally determined on model molecules. Details are thoroughly described in [18], [24] and

Calculations

We will report some calculations performed on polypeptides to show the reliability and the capabilities of our strategy. To accomplish this, we have compared ‘reference’ electron densities, namely charge distributions obtained by means of an exact calculation of the density matrix elements (i.e. using equation (1)), with those deriving from the new approximate expression. In order to have a quantitative comparison, we have considered the similarity index L(a, a′) introduced by Walker and Mezey

Conclusion

The transfer of ELMOs, previously determined on model molecules, is a way to assemble ab initio quality electron densities for very large molecules. However, the ELMOs non-orthogonality introduces a non-trivial task in the exact evaluation of the density matrix elements. In this letter, we have proposed an approximate expression for the computation of these elements, combining the extreme localization of the involved orbitals with the ‘Divide & Conquer’ philosophy. Furthermore, from preliminary

Acknowledgements

We thank Dr. Arianna Fornili, Dr. Monica Civera and Dr. Fausto Cargnoni for helpful discussions.

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A novel extremely localized molecular orbitals based technique for the one-electron density matrix computation

Abstract

Introduction

Section snippets

Theory

Calculations

Conclusion

Acknowledgements

Chem. Phys. Lett.

Comput. Chem.

J. Mol. Struct. – Theochem.

J. Mol. Struct. – Theochem.

Phys. Rev. Lett.

Rev. Mod. Phys.

Rev. Mod. Phys