Extension of frozen-density embedding theory for non-variational embedded wavefunctions

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Extension of frozen-density embedding theory for non-variational embedded wavefunctions

ZECH, Alexander, DREUW, Andreas, WESOLOWSKI, Tomasz Adam

Abstract

In the original formulation, frozen-density embedding theory [T. A. Wesolowski and A.

Warshel, J. Phys. Chem. 97, 8050–8053 (1993); T. A. Wesołowski, Phys. Rev. A 77, 012504 (2008)] concerns multi-level simulation methods in which variational methods are used to obtain the embedded NA-electron wavefunction. In this work, an implicit density functional for the total energy is constructed and used to derive a general expression for the total energy in methods in which the embedded NA electrons are treated non-variationally. The formula is exact within linear expansion in density perturbations. Illustrative numerical examples are provided

ZECH, Alexander, DREUW, Andreas, WESOLOWSKI, Tomasz Adam. Extension of

frozen-density embedding theory for non-variational embedded wavefunctions. Journal of Chemical Physics , 2019, vol. 150, no. 121101

DOI : 10.1063/1.5089233

Available at:

http://archive-ouverte.unige.ch/unige:140853

Disclaimer: layout of this document may differ from the published version.

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theory for non-variational embedded wavefunctions

Cite as: J. Chem. Phys. 150, 121101 (2019); https://doi.org/10.1063/1.5089233

Submitted: 17 January 2019 . Accepted: 03 March 2019 . Published Online: 29 March 2019 Alexander Zech , Andreas Dreuw , and Tomasz A. Wesolowski

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of Chemical Physics COMMUNICATION scitation.org/journal/jcp

Extension of frozen-density embedding theory for non-variational embedded wavefunctions

Cite as: J. Chem. Phys.150, 121101 (2019);doi: 10.1063/1.5089233 Submitted: 17 January 2019•Accepted: 3 March 2019•

Published Online: 29 March 2019

Alexander Zech,¹ Andreas Dreuw,² and Tomasz A. Wesolowski^1,a) AFFILIATIONS

1Department of Physical Chemistry, University of Geneva, 30, Quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland

2Interdisciplinary Center for Scientific Computing, Ruprecht-Karls University Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany

a)Electronic mail:[email protected]

ABSTRACT

In the original formulation, frozen-density embedding theory [T. A. Wesolowski and A. Warshel, J. Phys. Chem.97, 8050–8053 (1993);

T. A. Wesołowski, Phys. Rev. A77, 012504 (2008)] concerns multi-level simulation methods in which variational methods are used to obtain the embedded NA-electron wavefunction. In this work, an implicit density functional for the total energy is constructed and used to derive a general expression for the total energy in methods in which the embedded NAelectrons are treated non-variationally. The formula is exact within linear expansion in density perturbations. Illustrative numerical examples are provided.

Published under license by AIP Publishing.https://doi.org/10.1063/1.5089233

I. INTRODUCTION

Multi-scale simulation methods using a density-dependent embedding potential became popular in recent years.^1–7 Some of them use the formal framework of Frozen-Density Embedding Theory (FDET),^1,3 providing self-consistent expressions for the total-energy, the variationally obtained wavefunction representing embedded NAelectrons (ΨA), and the embedding potential. In particular, the key relation of FDET equating the total energy evaluated for the embedded wavefunction at a givenρB(⃗r)with the total energy expressed by means of the Hohenberg-Kohn density functional [see Eq.(1)below] holds only for embedded wavefunctions derived from variational methods such as Kohn-Sham, generalized- Kohn-Sham, Hartree-Fock, multiconfigurational self-consistent field, and configuration interaction (CI). For non-variational methods, the use of the FDET expression for the total energy (see, for instance, the expressions for the total energy given in Refs. 4–6 and8), represents anadditional approximation. The numerical sig- nificance of this approximation might be entangled with other approximations used in a particular method and vary from system to system. The formal link between FDET and the quantities obtained in non-variational calculations remains, however, to be established.

In the present work, we propose and derive a formal extension of FDET to non-variational methods to treat embedded NAelectrons.

II. FDET: EMBEDDED VARIATIONAL NA-ELECTRON WAVEFUNCTIONS

For a system comprising NABelectrons in an external potential v_AB(⃗r), the functionalE^EWFv_AB [ΨA,ρB]is defined to satisfy

minΨA

EÊWFv_AB [ΨA,ρB] =EÊWFv_AB [ΨôA,ρB] =E^HKv_AB[ρôA+ρB], (1) whereE^HKv_AB[ρ]is the Hohenberg-Kohn ground-state energy functional andρôA(⃗r) = ⟨ΨôA∣

N_A

∑

i

δ(⃗r− ⃗ri)∣Ψ^oA⟩.

By virtue of the second Hohenberg-Kohn theorem, Eq.(1)leads to

E^EWFv_AB [Ψ^oA,ρB] ≥E0, (2) whereE0=E^HKv_AB[ρ0]andρ0(⃗r)is the ground-state energy and density of the total system. Equality is reached for a large class of densities ρB(⃗r),

E^EWFv_AB [Ψ^oA,ρB] =E0 if ∀⃗r(ρ0(⃗r) >ρB(⃗r)). (3) Using conventional density functionals representing components of the total energy, and arbitrary partitioning of the external

J. Chem. Phys.150, 121101 (2019); doi: 10.1063/1.5089233 150, 121101-1

Published under license by AIP Publishing

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potentialvAB(⃗r) =vA(⃗r)+vB(⃗r), leads to the form ofE^EWFv_AB [ΨA,ρB] more suitable for further discussions,

E^EWFv_AB [ΨA,ρB] = ⟨ΨA∣HˆA∣ΨA⟩+VB[ρA]+JAB[ρA,ρB]+E^nadxcT[ρA,ρB] + ∆F[ρA]+E^HKv_B[ρB]+VA[ρB]+VN_AN_B, (4) where

VA[ρB] = ∫ vA(⃗r)ρB(⃗r)d⃗r, (5) VB[ρA] = ∫ vB(⃗r)ρA(⃗r)d⃗r, (6) JAB[ρA,ρB] = ∫ ∫ ρA(⃗r)ρB(⃗r^′)

∣⃗r− ⃗r^′∣

d⃗r^′d⃗r, (7) and VNANB is the interaction energy between the nuclei defining vA(⃗r) and vB(⃗r). The non-additive bi-functional E^nad_xcT[ρA,ρB] is related to the functionalsExc[ρ] andTs[ρ] defined in the constrained search formulation of the Kohn-Sham formalism,⁹

E^nadxcT[ρA,ρB] =Exc[ρA+ρB] −Exc[ρA] −Exc[ρB]

+Ts[ρA+ρB] −Ts[ρA] −Ts[ρB]. (8) The functional ∆F[ρ] on the other hand depends on the form of the wavefunction Ψ used in Eq. (1) and also is defined via the constrained search.³If ΨAis a single determinant (Φ), it reads

∆F[ρ] = min

ΦÐ→ρ

⟨Φ∣TˆNA+ ˆV_N^ee_A∣Φ⟩ −T[ρ] −Vee[ρ]

= ⟨Φô[ρ]∣TˆNA+ ˆV_Nêe_A∣Φô[ρ]⟩ −T[ρ] −Vee[ρ]

=Ec[ρ]. (9)

In the last line, we indicate that ∆F[ρ]is the correlation functional defined in constrained-search formulation of density functional theory.^9,10For Ψ of the full CI form, ∆F[ρ] =0 by definition.

Euler-Lagrange optimisation of ΨAleads to a Schrödinger-like equation,

(HˆA+ ˆv_emb)ΨA=λΨA, (10) where

v_emb[ρA,ρB,v_B](⃗r) =v_B(⃗r)+∫ ρB(⃗r^′)

∣⃗r− ⃗r^′∣ d⃗r^′

+v^nad_xcT[ρA,ρB](⃗r)+vF[ρA](⃗r), (11) withv^nad_xcT[ρA,ρB](⃗r), andvF[ρA](⃗r)being the first functional derivatives ofE^nad_xcT[ρ,ρB]and ∆F[ρ], respectively. The lowest energy solu- tion of Eq.(10)will be denoted as ΨÊL_A. Note that the energy is given not by the Lagrange multiplier λ but in Eq.(4). Any variational method can be used to obtain ΨÊLA and the corresponding density ρÊL_A(⃗r).

For a given density, ρ(⃗r), we can define the corresponding wavefunction as

min

ΨÐ→ρ(⃗r)

⟨Ψ∣ˆT+ ˆVee∣Ψ⟩ = ⟨Ψ^o[ρ]∣Tˆ+ ˆVee∣Ψ^o[ρ]⟩. (12) Up to a unitary transformation, it holds that

ΨÊL_A =Ψô[ρÊL_A]. (13) As a result, the second equality in Eq.(1)holds for variationally obtained wavefunctions ΨÊL_A and the corresponding density for any embedding potential. We should note that in Eq.(11), ˆv_embdepends onρA(⃗r). The first equality in Eq.(1)holds, therefore, only if ˆv_emb and ΨÊLA are self-consistent (ˆv_emb=v_emb[ρÊLA,ρB,vB](⃗r)). For lower- cost variational methods, this can be achieved in practice in an iterative procedure.¹¹

Owing to the fact that any NA electron wavefunction is an implicit (up to unitary transformation) functional of the density, the right-hand-side of Eq.(4)is an implicit functional ofρA(⃗r),

E^HKv_AB[ρA+ρB] =E^EWFv_AB [Ψ^o[ρA],ρB]

= ⟨Ψô[ρA]∣HÂ+ ˆv_emb[ρA,ρB,vB]∣Ψô[ρA]⟩

− ∫ ρA(⃗r)v_xcT^nad[ρA,ρB](⃗r)d⃗r

− ∫ ρA(⃗r)vF[ρA](⃗r)d⃗r+E^nad_xcT[ρA,ρB]+ ∆F[ρA] +E^HKv_B[ρB]+VA[ρB]+VN_AN_B. (14)

III. LINEAR EXPANSION OF THE FDET ENERGY FUNCTIONAL FOR EMBEDDED NON-VARIATIONAL WAVEFUNCTIONS

Let Ψ^nvarA and ρ^nvarA (⃗r) denote the embedded wavefunction and the corresponding density obtained from some non-variational method to solve Eq.(10). Contrary to the variational case,

Ψ^nvar_A ≠Ψ^o[ρ^nvar_A ]. (15) As a result, the FDET energy expression given in Eq.(14)cannot be applied unless the following constrained search is performed:

⟨Ψ^oA[ρ^nvarA ]∣ˆT+ ˆVee∣Ψ^oA[ρ^nvarA ]⟩ = min

ΨAÐ→ρ^nvar_A

⟨ΨA∣Tˆ+ ˆVee∣ΨA⟩. (16) Performing such a search is, however, impractical. More impor- tantly, for methods in which the total energy is not evaluated as the expectation value of the Hamiltonian, Eq.(4)cannot be used. More- over, the density is either not available in a straightforward manner (in coupled-cluster type of approaches, for instance) or it is available in the lower order than energy [Møller-Plesset perturbation theory (MP), for instance].

In the following, we exploit the fact that the right-hand side of Eq. (14) represents an implicit density functional, which can be expanded in a series at ρ^ELA(⃗r). For small differences,

∆ρ(⃗r) =ρ^nvarA (⃗r) −ρ^ELA(⃗r)retaining the linear terms leads to

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E^HKv_AB[ρÊLA + ∆ρ+ρB] =EÊWFv_AB [Ψô[ρÊLA + ∆ρ],ρB]

= ⟨Ψ[ρÊLA]∣HÂ+ ˆv_emb[ρÊLA,ρB,vB]∣Ψ[ρÊLA]⟩ − ∫ ρÊLA(⃗r)v^nad_xcT[ρÊLA,ρB](⃗r)d⃗r

− ∫ ρÊLA(⃗r)vF[ρÊLA](⃗r)d⃗r+E^nadxcT[ρÊLA,ρB]+ ∆F[ρÊLA]+E^HKv_B[ρB]+VA[ρB]+VNANB

− ∫ ρ^EL_A(⃗r) ∫ δv^nad_xcT[ρ,ρB](⃗r)

δρ ∣

ρ=ρ^EL_A(⃗r^′)

∆ρ(⃗r^′)d⃗rd⃗r^′− ∫ ρ^EL_A(⃗r) ∫ δvF[ρ](⃗r) δρ ∣

ρ=ρ^EL_A(⃗r^′)

∆ρ(⃗r^′)d⃗rd⃗r^′+O(∆ρ²). (17)

Besides the last two terms, other linear terms in ∆ρ(⃗r)do not sur- vive ifρ^EL_A(⃗r)is obtained variationallyandwith the self-consistent embedding potential.

Equation(17)is our principal result. It is exact up to first order in ∆ρ(⃗r)provided the embedded wavefunction is obtained variationally with the self-consistent embedding potential. Equation(17) can be used within various non-variational treatments of the embedded NA electrons and approximations to the explicit density functionals needed in practice. In a method, for which a correction to the energy obtained from variational calculations (correlation energy from Møller-Plesset or coupled-cluster calculations, for instance) is available, it can be expected to provide a good approximation to ∆F[ρ^EL_A]. If a method yields also ∆ρ(⃗r)(at lower order than the energy in MP calculations, for instance), it can be used in the last two terms.

IV. NUMERICAL EXAMPLE

This section illustrates the usefulness of Eq. (17) as a basis for approximate methods. For several weakly bound intermolecular complexes, the total energies are obtained using the following choice of the simplest approximations in Eq.(17). ∆F[ρ^ELA]is approximated by means of the correlation energy from second-order Møller-Plesset perturbation theory (MP2). ForE^nad_xcT[ρ^EL_A,ρB], local density approximation is used (decomposable approximations using Dirac-Slater functional for exchange energy,^12,13the parameterisa- tion of the Ceperley-Alder correlation energy¹⁴ by Voskoet al.,¹⁵ and Thomas-Fermi functional for kinetic energy.^16,17 The use of local density approximation for all relevant functionals was moti- vated by expected cancellation of errors (sharing the same approximation for the averaged exchange-correlation hole) confirmed in numerical tests on a representative set of pairs of densitiesρA(⃗r) andρB(⃗r).^18,19In the absence of well-tested approximations for the second functional derivatives, the ∆ρ(⃗r)-dependent contributions to energy are neglected. Preliminary numerical results using local density approximation indicate that these terms are indeed negligi- ble. ConcerningρB(⃗r), it was optimised using the iterative freeze- and-thaw procedure.²⁰In view of practical applications, we discuss below not the total energies obtained from the defined above approximated version of Eq.(17)but the interaction energies. The FDET interaction energies obtained in this way are compared to conventional counterpoise corrected MP2 interaction energies. Optimisa- tion ofρB(⃗r)is made in order to attribute the errors in the FDET interaction energiesonlyto approximations made in Eq.(17). To

FIG. 1.E^FDET_int and its components together with counterpoise corrected MP2 interaction energies (E^MP2_int shown as black lines):Eelst = VB[ρ^EL_A]+VA[ρB] +JAB[ρ^EL_A,ρB]+VN_AN_B,∆E^MP2_int =E^FDET_int −E^MP2_int ,∆E⁽²⁾_A =E⁽²⁾_A,emb−E⁽²⁾_A .

J. Chem. Phys.150, 121101 (2019); doi: 10.1063/1.5089233 150, 121101-3

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FIG. 2. FDET interaction energy and its components. See also caption toFig. 1.

validate numerical robustness of Eq.(17), the reported FDET energies are evaluated using three atomic basis sets (cc-pVDZ, cc-pVTZ, and cc-pVQZ²¹) in two types of multi-center expansion: either localised on individual molecules (monomer expansion,ME) or on all atoms in the system (supermolecular expansion,SE). The FDET interaction energies are calculated as

E^FDET,MP2_int = ⟨ΦÊL∣HÂ∣ΦÊL⟩ − ⟨Φ^free∣HÂ∣Φ^free⟩+VB[ρÊLA]+VA[ρB] +JAB[ρÊLA,ρB]+VN_AN_B+E^nadxc [ρÊLA,ρB]+Ts^nad[ρÊLA,ρB] +(E⁽²⁾_A,emb−E⁽²⁾_A )+(EêmbB −E_B^free), (18) whereEêmbB is the total MP2 energy of subsystem B used asE^HKv_B[ρB] in Eq.(17)andE_B^freeis the total MP2 energy of the isolated subsystem

B. Note that ifρB(⃗r)is not optimized, the difference between these two energies is zero.E⁽²⁾_A,embandE⁽²⁾_A denote the correlation energies obtained from second-order perturbation theory using embedded and environment-free reference states, respectively.

The results for four representative intermolecular complexes at their equilibrium geometries are reported inFigs. 1and2. FDET energies show a remarkable agreement with the reference interaction energies for each basis set. Rather small variation of the FDET interaction energy and its components with the used basis set indicate numerical robustness of the method. Note also that the contributions to the energy, which are approximated using explicit functionals (E^nadxc [ρÊLA,ρB]andT^nads [ρÊLA ,ρB]), are numerically significant, opposite in sign, and of similar magnitude as the classical electro- static interactions. This provides an additional proof of the good cancellation of errors inE^nad(LDA)_xc [ρÊL_A,ρB]andT_s^nad(LDA)[ρÊL_A,ρB]. If other approximations for these functionals would be considered, the balance in their quality is more important than improving the accu- racy of only one of them. The cause of this remarkable performance is the object of our current research and will be reported in a full publication.

ACKNOWLEDGMENTS

This research was supported by a grant from the Swiss National Science Foundation (Grant No. 200020-172532).

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J. Chem. Phys.150, 121101 (2019); doi: 10.1063/1.5089233 150, 121101-5