A Variational Approach to London Dispersion Interactions Without

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A Variational Approach to London Dispersion Interactions Without Density Distortion Derk Pieter Kooi, and Paola Gori-Giorgi J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b00469 • Publication Date (Web): 13 Mar 2019 Downloaded from http://pubs.acs.org on March 13, 2019

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The Journal of Physical Chemistry Letters

A Variational Approach to London Dispersion Interactions Without Density Distortion Derk P. Kooi and Paola Gori-Giorgi∗ Department of Theoretical Chemistry and Amsterdam Center for Multiscale Modeling, Faculty of Science, Vrije Universiteit Amsterdam, The Netherlands E-mail: [email protected]

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The Journal of Physical Chemistry Letters 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract We introduce a class of variational wavefunctions that capture the long-range interaction between neutral systems (atoms and molecules) without changing the diagonal of the density matrix of each monomer. The corresponding energy optimization yields explicit expressions for the dispersion coefficients in terms of the ground-state pair densities of the isolated systems, providing a clean theoretical framework to build new approximations in several contexts. As the individual monomer densities are kept fixed, we can also unambiguously assess the effect of the density distortion on London dispersion interactions: for example, we obtain virtually exact dispersion coefficients between two hydrogen atoms up to C10 , and relative errors below 0.2% in other simple cases.

Graphical TOC Entry P (r1 , r2 ) ⇢(r1 )⇢(r2 )

Z

⇢(r1 )⇢(r2 )[1 + J(r1 , r2 )]

⇢(r1 ) dr2

Z

dr2

Keywords dispersion, van der Waals, density functional theory

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Within the Born-Oppenheimer approximation, non-relativistic quantum mechanics of Coulomb systems (atoms, molecules) predicts that everything is attracted to everything, since it is always possible to find a relative orientation that lowers the total energy of two neutral molecules separated by a large distance R by at least a term −C6 R−6 , with C6 > 0. 1,2 This universal attractive interaction between quantum Coulomb systems, often called London dispersion interaction, is very small compared to the energy scale of a typical chemical (covalent) bond. Yet, it is omnipresent in chemistry, biology, and physics, determining protein folding, the structure of DNA, the physics of layered materials, just to name a few examples. An accurate, efficient, and fully non-empirical treatment of these forces remains an open challenge in many respects, and it is the focus of several on-going research efforts. 3–9 The physical origin of London dispersion interactions is often discussed in terms of “coupling between instantaneous dipoles”, 2,10 which are actually correlations in the interfragment part of the pair density (the diagonal of the two-body reduced density matrix), as illustrated, e.g., in. 11 The textbook treatment is based on perturbation theory, 2,10,12 in which excitations on both monomers are coupled, which allows to rewrite the dispersion coefficients in terms of dynamical polarizabilities of the individual systems. In this Letter, inspired by the work of Lieb and Thirring (LT), 1 we attack the dispersion problem from a variational perspective, using only ground-state properties of the monomers. We construct a class of variational wavefunctions for the long-range interaction between neutral Coulomb systems that are explicitly forbidden to deform the diagonal of the full many-body spatial density matrix (and thus the density) of each individual monomer. This constraint has several advantages: first of all, it is conceptually appealing, as dispersion becomes a monomer-monomer interaction accompanied by a change in kinetic energy only, since all the remaining intra monomer interactions are kept unchanged. Secondly, it highly simplifies the optimization procedure, which can be implemented in a very efficient way and leads to expressions for the dispersion coefficients in terms of the ground-state pair densities of the individual monomers, giving a neat framework to build new approximations. For example, in a density functional theory (DFT)

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setting, one can use approximations for the exchange-correlation holes of the fragments, providing justification and a route for improving models such as the exchange-hole dipole moment (XDM). 13 It also provides a prescription to obtain atomic dispersion coefficients inside molecules and solids, which are a crucial ingredient in the construction of atomistic force fields and dispersion corrections to DFT calculations (DFT+D). One could easily object that this construction cannot give the exact answer. In a widely quoted paper, Feynman 14 wrote that van der Waals’ forces arise from charge distributions with higher concentration between the nuclei, with a permanent dipole moment of order R−7 being induced on each atom, such that “each nucleus is attracted by the distorted charge distribution of its own electrons”. 14 Feynman’s conclusion was proven by using the electrostatic Helmann-Feynman theorem for both atoms and molecules, 15 and has been confirmed with various accurate calculations. 16,17 Therefore we know that a small density distortion must be there. Nonetheless, it has been observed several times that very accurate dispersion coefficients can be obtained with methods that describe very poorly this density distortion, including cases in which the electrostratic Helmann-Feynman theorem gives zero force at order R−7 . 18 The issue has been often debated in the literature, and there is the general feeling that the density distortion, which actually requires expensive calculations to be accurately captured, should have very little influence on dispersion energetics, which is our goal here. Moreover, our construction provides (if combined with accurate pair densities for the monomers), a rigorous upper bound to the best possible variational interaction energy when the density distortion is explicitly forbidden, allowing us to assess its importance in an unambiguous manner, which is per se also conceptually interesting. For illustrative purposes, we discuss first the case of two systems A and B, each with NA = NB = 1 electron, generalizing it right after to the many-electron case. The two atoms are separated by a large distance R, and our wavefunction reads p B 1 + JR (r1 , r2 ), (r ) ΨR (r1 , r2 ) = ΨA (r )Ψ 1 2 0 0 4


(1)

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where r1 and r2 are electronic coordinates centered in nucleus A and nucleus B, respecA(B)

tively, Ψ0

(r) is the ground-state wavefunction of system A (or B) alone, and JR , with

|JR (r1 , r2 )| 1, preserves the unperturbed densities at all orders in R−1 via the constraints Z

Z

2 dr1 |ΨA 0 (r1 )| JR (r1 , r2 ) = 0

∀ r2

(2)

2 dr2 |ΨB 0 (r2 )| JR (r1 , r2 ) = 0

∀ r1 ,

(3)

B which are imposed with an expansion in terms of one-electron functions bA i (r) and bi (r) and

variational parameters cij,R ,

JR (r1 , r2 ) =

X

B cij,R bA i (r1 )bj (r2 ),

(4)

Z

(5)

ij

where bA i (r) A(B)

(and similarly for bB j ), with ρ0

=

fiA (r)

−

A(B)

(r) = |Ψ0

ρA 0 (r) A f (r)dr NA i A(B)

(r)|2 . The fi

(r)’s can be chosen in differ-

ent ways, with the choice influencing the convergence. Our wavefunction does not include antisymmetrization between the electron on A and the one on B, which affects the asymptotic (large R) interaction energy only with terms that vanish exponentially with R. This wavefunction correlates the positions of the two electrons in the two different atoms without changing their one-electron densities. ˆ =H ˆ A (r1 )+ H ˆ B (r2 )+ H ˆ int (r1 , r2 ), We now consider the full hamiltonian of the problem, H ˆ A and H ˆ B are the hamiltonians of each isolated atom and H ˆ int contains all the where H interactions between the particles in A (electron and nucleus) with those in B, and evaluate its expectation value on our wavefunction, focussing on the interaction energy Eint = E0AB − E0A − E0B . Thanks to the density constraint of Eqs. (2)-(3), there is no change in the expectation value of the nuclear-electron potential, implying that the interaction energy is entirely determined by the change in kinetic energy Tˆ with respect to the one of the isolated

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atoms, ∆ T ({cij,R }) = hΨR |Tˆ|ΨR i − T0A − T0B ,

(6)

ˆ int , which splits automatically into the sum of an electrostatic plus the expectation value of H part Wint,0 (R) and the correlation part Wint,c ({cij,R }, R), B ˆ A B Wint,0 (R) = hΨA 0 Ψ0 |Hint |Ψ0 Ψ0 i , Z ˆ int |ΨA |2 |ΨB |2 JR dr1 dr2 , Wint,c ({cij,R }, R) = H 0 0

(7)

Eint (R) = ∆ T ({cij,R }) + Wint,0 (R) + Wint,c ({cij,R }, R).

(9)

(8)

ˆ int , As usual, 2 we perform a multipolar expansion of H ˆ ˆ ˆ ˆ ˆ ˆ ˆ int = Hdd + Hdq + Hqd + Hqq + Hod + Hdo + ..., H R3 R4 R5

(10)

where d stands for dipole, q for quadrupole, o for octopole, etc. The variational problem then takes a simplified form, as detailed in the supplementary material, 19 with each order R−2n in the energy having its optimal cij,R = cij R−n . The physics described by this wavefunction can be understood by looking at the leading order: our wavefunction gives a ∆ T ({cij }) which is positive (as it costs kinetic energy to correlate the electrons) and quadratic in the variational parameters cij , while Wint,c ({cij }, R) is linear in the cij , negative (as correlation occurs exactly to lower the interaction term), and has a prefactor R−3 from (10). This guarantees that there are always optimal variational parameters cij , all proportional to R−3 , which minimize Eint , whose minimum becomes the familiar −C6 /R6 . Notice that ∆T is quadratic in the cij only by virtue of the density constraint. This structure repeats at each order in 1/R. In LT 1 and subsequent works in a similar spirit, 20,21 the energy cost to correlate the electrons, quadratic in the variational parameters, comes from the full hamiltonian of the monomers, while with our construction we force it to be of kinetic energy origin only. Solving the variational equations leads to explicit expressions for all the even dispersion 6


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coefficients. For example, C6 is given by 1 C6 = wT L−1 w, 2

(11)

where the vector w has contracted indices I = ij,

wij =

X

he

e

Z

2 A e|ΨA 0 (r)| bi (r)dr

Z

2 B e|ΨB 0 (r)| bj (r)dr,

(12)

with e = x, y, z, and he = (1, 1, −2) when the bond is along the z-axis. The matrix L has two contracted indices I = ij and K = kl,

Lij,kl =

1 A B A B τik Sjl + Sik τjl , 4

(13)

with

τijA SijA

Z

= =

Z

2 A A |ΨA 0 (r)| ∇bi (r) · ∇bj (r) dr,

(14)

2 A A |ΨA 0 (r)| bi (r)bj (r) dr,

(15)

and similarly for B. Analogous expressions hold for all the even dispersion coefficients (see 19 ). A(B)

We have started with a very simple choice for the fi

(r)’s (just powers of the distance

r from the nucleus times spherical harmonics 19 ), finding that the convergence is already fast, as shown in Fig. 1 for the case of C6 for two H atoms. The values of C6 , C8 and C10 for the same case are reported in Table 1, where they are compared with reference data from Ref. 22. In the supplementary material, 19 we also report even second-order coefficients up to C30 , which also agree within numerical accuracy with those of Ref. 22. From Table 1 we see that our wavefunction yields essentially the exact lowering in energy due to dispersion without any distortion of the density of each individual atom. We now turn to the many-electron case, by considering two systems with NA and NB

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��

Figure 1: Relative error on C6 between two H atoms with increasing number of terms in Eq. (4) relative to the reference result. 23 Table 1: Dispersion coefficients C6 , C8 and C10 between two H atoms computed with 30 terms in the variatonal expression of Eqs. (1)-(15), compared to the reference data of Thakkar 22 and Masili, et al. 23 See also supplementary material. 19

C6 C8 C10

Thakkar 22 6.4990267054058405 × 100 1.2439908358362235 × 102 3.2858284149674217 × 103

Masili et al. 23 6.4990267054058393 × 100 n.a. n.a.

This work (30 terms) 6.4990267054058393 × 100 1.2439908358362234 × 102 3.2858284149674217 × 103

electrons, for which our wavefunction reads

ΨR (xA , xB ) =

B ΨA 0 (xA )Ψ0 (xB )

s

1+

X

JR (ri , rj ),

(16)

i∈A,j∈B

where xA(B) indicates all the electronic coordinates (including spins) of the electrons in A(B)

system A (or B), and, again, Ψ0

is the ground state wavefunction of system A (or B)

alone. We choose JR such that the diagonal of the full many-body density matrix of each system is not allowed to deviate from its ground-state expression: Z

Z

2 |ΨR (xA , xB )|2 dxA = |ΨB 0 (xB )|

∀ xB

(17)

2 |ΨR (xA , xB )|2 dxB = |ΨA 0 (xA )|

∀ xA ,

(18)

8


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which is enforced by choosing for JR the same form as Eq. (4), i.e., by imposing that Z Z A(B)

with ρ0

ρA 0 (riA )JR (riA , rjB )driA = 0

∀ rjB

(19)

ρB 0 (rjB )JR (riA , rjB )drjB = 0

∀ riA ,

(20)

the ground-state one-electron densities of the two systems. Again, it is only the

interfragment pair density and the off-diagonal elements of the first-order reduced density matrix that are allowed to change to lower the energy, implying that also in the manyelectron case the interaction energy is given by Eq. (9), since the expectation value of the electron-electron interaction inside each monomer is kept unchanged by virtue of Eqs. (17)(18). The variational minimization of the interaction energy is similar to the N = 2 case. For example, C6 takes the same form as Eq. (11), where now w and L read

wij =

X

A B B he (dA i + Di )(dj + Dj ),

(21)

e=x,y,z

Lij,kl =

1 A B A τik (Sjl + PjlB ) + (Sik + PikA )τjlB , 4

(22)

where (with the same expressions for B)

dA i

Z

A dr1A ρA 0 (r1A )bi (r1A )e1A , Z Z A Di = dr1A dr2A P0A (r1A , r2A )bA i (r2A )e1A , Z Z A PijA = dr1A dr2A P0A (r1A , r2A )bA i (r1A )bj (r2A ), A(B)

and where τij

=

A(B)

and Sij

(23) (24) (25)

are defined in Eqs. (14)-(15). We see that the dispersion coeffi-

cients Cn for the many-electron case become explicit functionals of the spin-summed groundstate pair densities P0A (r1A , r2A ) and P0B (r1B , r2B ) of the two monomers, Cn = Cn [P0A , P0B ]. The variational solution for the optimal cij can also be turned into a Sylvester equation, which can be solved in an efficient way. 19,24 9



Being derived from a wavefunction, the expression for the interaction energy is variational (i.e., it provides a lower bound to the exact C6 ) as long as we use exact (or very accurate) pair A(B)

densities for the monomers. As an example, with the same simple choice for the fi

(r) used

for two H atoms, we have computed C6 for He-He using the accurate variational wavefunction for the He atom of Freund, Huxtable and Morgan 25 (see also Ref. 26). The convergence is again fast, very similar to the one of Fig. 1, yielding C6 = 1.458440 with an error of 0.17% with respect to the accurate value 1.460978 of Ref. 27. For the He-H case we have a similar error of 0.15% with respect to Ref. 27. Whether these residual errors are due to the reduced variational freedom of our wavefunction or to inaccuracies in the multiple moment integrals of the correlated He pair density of Ref. 25 is for now an open question, but we see that the results can be rather accurate, at a computational cost which is determined by the individual monomer calculations. This opens many possibilites, as our expressions can be coupled with any wavefunction method for the monomers, provided it gives access to the second-order A(B)

reduced density matrices and their integrals with the fi

(r), whose optimal choice needs

to be assessed in a systematic way. 10 ��-�� -��-�� 8 C6

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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● ■ ◆ ▲

● ■ ◆ ▲

6 ��

��-��-�� 5

● ■ ◆ ▲

● ■ ◆ ▲

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●

aug-cc-pVDZ

■

aug-cc-pVTZ

◆

aug-cc-pVQZ

▲

aug-cc-pV5Z

10 nmax

15

20

A(B)

Figure 2: Convergence of the C6 dispersion coefficient with the number of functions fi (r) for Ne-Ne using Hartree-Fock pair densities of the individual atoms in different basis sets. HF calculations were carried out using Gaussian 16 28 and the arbitrary order multipole moments were calculated using HORTON 3. 29 Accurate value (thick line) from Ref. 30. Values for vdW-DF-04/10 and VV-09 are from Ref. 31, while the value for XDM was taken from Ref. 13. vdW-DF-09 was not included, since it is parametrized based on this dispersion coefficient. We can of course use pair densities from lower level theories, which opens another whole 10


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realm of possibilities, bearing in mind that the expressions will be no longer variational for the interaction energy. For example, in Fig. 2 we show the result for C6 for Ne-Ne using the Hartree-Fock (HF) pair density, including the basis set dependence. We see that the error is around 5%, as HF is a good approximation for the Ne atom, and that we are below the exact interaction energy. It is clear that a detailed study of the basis set dependence A(B)

is also needed, which should be coupled to an optimal choice for the fi

(r). The use of

Kohn-Sham orbitals from different approximate functionals to determine the exchange pair densities is also worth to be investigated. Within DFT, one could make even simpler approximations for the exchange-correlation A(B)

holes hxc

of the monomers, transforming the dispersion coefficients into density functionals,

A B B A B Cn [P0A , P0B ] = Cn [ρA 0 , hxc , ρ0 , hxc ] → Cn [ρ0 , ρ0 ].

(26)

For example, the Becke-Roussel 32 exchange-hole model could be used to estimate higher moments of the exchange pair density, 33 in a spirit similar to the XDM model. 13 The main difference is that now the polarizability of the monomers is no longer needed, as the expression for the dispersion coefficients only requires the exchange-correlation holes. An important ingredient in atomistic force fields models and DFT+D are the dispersion coefficients Cn,ab between pairs of atoms a b, with a in system A and b in system B, which are usually computed from atomic polarizabilities. Our variational wavefunction can provide B a framework to define and compute them. One should obtain atomic volumes ΩA a and Ωb

using one of the various definitions (e.g. Hirshfeld partitioning 34 ), and then obtain the pair dispersion coefficients using, order by order, the energy density coming from our wavefunction with the cij optimized for the AB interaction, restricting the integration on the atomic

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volumes. For example for C6 (with J below being the optimized factor at leading order), C6,ab

1 = 2

Z

ΩA a

dr1A Z

Z

ΩB b

B ˆ dr1B Hdd (r1A , r1B ) ρA 0 (r1A )ρ0 (r1B )

× J(r1A , r1B ) + dr2A P0A (r1A , r2A )ρB 0 (r1B )J(r2A , r1B ) Z B + dr2B ρA 0 (r1A )P0 (r1B , r2B )J(r1A , r2B ) Z A B + dr2B dr2A P0 (r1A , r2A )P0 (r1B , r2B )J(r2A , r2B )

(27)

and similarly for the higher order coefficients. In conclusion, the class of variational wavefunctions we have introduced here provides a neat theoretical framework to build new approximations to tackle dispersion interactions, which are reduced to a competition between kinetic energy and monomer-monomer interaction only. Although this class of variational wavefunctions violates the virial theorem (similarly to second-order perturbation theory), and although we know that for the exact wavefunction the density must be distorted, the results can be very accurate (or even exact), with a computational efficiency that is promising for the many-electron case. Moreover, we know what is being approximated, so that one could also devise a scheme in which the monomer (pair) densities are relaxed in a second step, or perturbatively, if this is needed. The realm of possibilities open is vast, ranging from the use of various wavefunction methods to compute the monomer pair densities or approximations thereof, to the definition and calculation of atomic-pair dispersion coefficients. The influence of the level of theory on the A(B)

calculation of the monomer pair densities, the basis set, and different choices for the fi

(r)

need all to be investigated. We have also shown that exact dispersion coefficients up to C10 between two hydrogen atoms can be obtained without introducing any density distortion, which is interesting by itself. The framework can be also generalised to many monomers and to describe the dispersive part of the interaction between charged systems.

12


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Acknowledgement Financial support from European Research Council under H2020/ERC Consolidator Grant corr-DFT (Grant Number 648932) and the Netherlands Organisation for Scientific Research under Vici grant 724.017.001 is acknowledged.

Supporting Information Available The Supporting Information is available free of charge on the ACS Publications website at DOI:.

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A Variational Approach to London Dispersion Interactions Without

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