Potential Functional Embedding Theory with an Improved KohnâSham

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Quantum Electronic Structure

Potential Functional Embedding Theory with an Improved Kohn-Sham Inversion Algorithm Qi Ou, and Emily A. Carter J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00717 • Publication Date (Web): 14 Sep 2018 Downloaded from http://pubs.acs.org on September 18, 2018

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Journal of Chemical Theory and Computation

Potential Functional Embedding Theory with an Improved Kohn-Sham Inversion Algorithm Qi Ou1 and Emily A. Carter2,* 1

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ

08544 2

School of Engineering and Applied Science, Princeton University, Princeton, NJ 08544

Abstract Potential functional embedding theory (PFET) is a rigorous theory that can yield a unique, selfconsistent embedding potential shared by different subsystems treated at different levels of theory. Application of PFET has been limited by the time-consuming and sometimes unstable optimized effective potential (OEP) procedure. Here we improve the performance of PFET by replacing the OEP algorithm with a new method to reconstruct the effective Kohn-Sham (KS) potential. We propose a direct, efficient KS inversion algorithm to solve for the effective KS potential, and then employ the resulting algorithm in PFET. We benchmark our KS inversion algorithm against the recently reported modified Ryabinkin-Kohut-Staroverov (mRKS) procedure. Numerical examples show that, with sufficiently large basis sets, our KS inversion algorithm generates almost as accurate results as the mRKS procedure does, except in the vicinity of atomic nuclei, and that it requires less computational time. Three types of chemical *

Email: [email protected]

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interactions then were tested using the new KS inversion algorithm in PFET; the energetics computed from the updated formalism compare well to benchmarks.

1 Introduction Obtaining accurate interaction energies has long been one of the major challenges in molecular systems and condensed-matter physics.1–6 The Coulomb interaction between electrons makes it exponentially difficult to diagonalize an -body Hamiltonian exactly. High-level correlated wavefunction (CW) methods, such as complete-active-space self-consistent field (CASSCF) theory,7 multi-reference configuration interaction (MRCI) theory,8,9 and coupled-cluster theory,10 were developed to provide an accurate electronic structure determination. However, the expensive computational cost of these methods has limited their application to relatively small numbers of atoms, while larger numbers of atoms are usually treated with mean-field theories such as density functional theory (DFT), instead.11–14 Embedded correlated wavefunction (ECW) methods have been proposed to overcome this limitation by incorporating a CW-level embedded region (subsystem I) into, e.g., a DFTlevel environment (subsystem II).15–17 Within the framework of orbital-free embedding theory, the interaction between the surroundings and embedded region is comprehensively described by an embedding potential.18–20 Two recently proposed embedding schemes in our group, namely, the density functional embedding theory (DFET)21 and potential functional embedding theory (PFET),22 assume a common embedding potential shared by all subsystems that uniquely determines the density partitioning of the total system. Unlike DFET, which computes the mutual polarization of the subsystems at the DFT level, PFET provides a fully self-consistent hybrid CW/DFT calculation that enables the interactions between different subsystems to go


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beyond the DFT level.23,24 Nevertheless, PFET unavoidably is much more time-consuming compared to DFET due to the fact that, at each self-consistent iteration, the kinetic potentials need to be evaluated explicitly with an effective KS potential for both the cluster and the total system.22 Successful implementation of an efficient and stable algorithm to solve for the effective KS potential thus is essential in PFET. The original PFET recipe employed the Wu-Yang OEP algorithm, which requires auxiliary basis sets to expand the yet undetermined effective potential.25 The choice of an auxiliary basis set is non-trivial and can sometimes lead to unphysical results.26–28 The Wu-Yang OEP procedure in the original PFET recipe was implemented with plane-wave basis sets, which are more complete than the atomic orbital (AO) basis sets and therefore make the Wu-Yang OEP procedure more stable. However, generating an OEP for finite systems using plane-wave basis sets is very inefficient, so an AO basis set is preferred for such systems. A recently reported algorithm from Staroverov and co-workers, namely, the modified Ryabinkin-Kohut-Staroverov (mRKS) procedure, provides a very satisfactory way to solve for the effective KS potential.29–33 In the mRKS procedure, one derives two expressions for the local electron energy balance, one from the iterative KS equation and the other from the wavefunction-based Schrödinger equation. The effective exchange-correlation (XC) potential is obtained by subtracting one expression from the other under the assumption that the final, converged KS- and wavefunction-based densities are equal.29,31 The effective KS potential for the wavefunction-based density then is constructed by summing over the effective XC potential and the corresponding Coulomb and external potentials. Numerical examples proved that the mRKS procedure is robust for almost all Gaussian basis sets and can construct unambiguous, nearly exact effective potentials at the same low cost as the Krieger-Li-Iafrate34 and Becke-Johnson35 potentials.29


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Despite its promising advantages, the mRKS procedure has two limitations that may narrow its application. First, it requires the direct construction of the Slater potential from the wavefunction-based calculation, which is still a computationally expensive, rate-limiting step.29 Second, and more importantly, the mRKS procedure requires the molecular orbitals and twoelectron reduced density matrix (2RDM) of the target system to compute the effective potential31,33 (although for the Hartree-Fock (HF) density, it only requires the one-electron reduced density matrix (1RDM) and the orbital information29). The mRKS procedure therefore cannot be applied to cases where the orbitals and 2RDM are not available, such as the total density in PFET, which is a summation of subsystem densities. To overcome these two limitations, here we propose a direct and efficient KS inversion algorithm to construct the effective KS potential for PFET. In this algorithm, we directly invert the KS equation for the target density and iteratively solve the KS equation until the density is converged. Our algorithm does not require the wavefunction, 2RDM, or the construction of the Slater potential, making it universally applicable to any case where the density is v-representable. Herein, we apply the resulting effective KS potential to the PFET algorithm, and investigate different types of interactions to test its transferability. This article is organized as follows. In Section 2, we briefly review the basic formulation of PFET and derive our KS inversion algorithm, followed by a brief introduction of the mRKS procedure. Section 3 provides our results with the updated formalism In Section 3.1, we compare the performance of the mRKS procedure and our KS inversion algorithm using three numerical examples. In Section 3.2, we investigate three types of chemical interactions with the updated PFET formalism. We conclude in Section 4.


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2 Theory 2.1 PFET In this subsection, we briefly introduce the basic concepts and central equations in PFET (see Ref. 6 for a complete, detailed derivation). As a member of the orbital-free embedding theory family, PFET partitions only the electron density of the total system, instead of the wavefunction or the density matrix, into an arbitrary number of different subsystems; these subsystems are often just a cluster of atoms (treated at the CW level) and its environment (treated at the DFT level).22–24 The interaction between the cluster and its environment is given by the embedding potential shared by the subsystems. The uniqueness of the embedding potential, as well as the density partition, was proven in Ref. 22. In PFET, the total energy is calculated as a functional of this unique embedding potential. For the case of only two subsystems, the total energy is = + +

(1)

where and denote the energy of the embedded cluster (I) and the embedded environment (II), respectively, and denotes the interaction energy between the two subsystems. is a functional of both subsystem densities, including the Hartree, XC, kinetic, and electron-ion interaction energies = + , + , + ∑ ∑, − ∑ , #, $ = %, %%

(2)

where = ∑ − ∑

(3)

, = ∑ − ∑

(4)

, = ∑ − ∑

(5)


5


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Here, , , and are the Hartree, XC, and kinetic energies, respectively, for a given density . The last term in Eqn. (2) is subtracted to avoid double counting, since the same embedding energy is included already in the embedded subsystem calculations. The total energy is minimized with respect to the embedding potential using optimization algorithms such as L-BFGS-B.36,37 We evaluate the functional derivative of the total energy with respect to the embedding potential using the second-order finite-difference approximation, in which the difference between subsystem densities evaluated in the presence of two different embedding potentials approximates the derivative &'()( *+,-

&*+,-

.

= ∑9 , /0 1 2 + 3

&'45( 64

&64

7 − 2 − 3

&'45( 64

&64

78

(6)

We evaluate the functional derivative of the interaction energy with respect to the subsystem density as :

= ;, + , +

Potential Functional Embedding Theory with an Improved KohnâSham

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Potential Functional Embedding Theory with an Improved KohnâSham