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A Generally Applicable Atomic-Charge Dependent London Dispersion

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A Generally Applicable Atomic-Charge Dependent London Dispersion A generally applicable atomic-charge dependent London dispersion correction Eike Caldeweyher,1 Sebastian Ehlert,1 Andreas Hansen,1 Hagen Neugebauer,1 Sebastian Spicher,1 Christoph Bannwarth,1, 2 and Stefan Grimme1, a) 1)Mulliken Center for Theoretical Chemistry, Institut f¨urPhysikalische und Theoretische Chemie der Universit¨atBonn Beringstr. 4, D-53115 Bonn, Germany 2)Department of Chemistry, Stanford University, Stanford, CA 94305, United States of America (Dated: 25 January 2019) The D4 model is presented for the accurate computation of London dispersion interactions in density func- tional theory approximations (DFT-D4) and generally for atomistic modeling methods. In this successor to the DFT-D3 model, the atomic coordination-dependent dipole polarizabilities are scaled based on atomic partial charges which can be taken from various sources. For this purpose, a new charge-dependent parameter- economic scaling function is designed. Classical charges are obtained from an atomic electronegativity equi- libration procedure for which efficient analytical derivatives are developed. A numerical Casimir-Polder integration of the atom-in-molecule dynamic polarizabilities yields charge- and geometry-dependent dipole- dipole dispersion coefficients. Similar to the D3 model, the dynamic polarizabilities are pre-computed by time-dependent DFT and elements up to radon are covered. For a benchmark set of 1225 dispersion coeffi- cients, the D4 model achieves an unprecedented accuracy with a mean relative deviation of 3.8% compared to 4.7% for D3. In addition to the two-body part, three-body effects are described by an Axilrod-Teller-Muto term. A common many-body dispersion expansion was extensively tested and an energy correction based on D4 polarizabilities is found to be advantageous for some larger systems. Becke-Johnson-type damping parameters for DFT-D4 are determined for more than 60 common functionals. For various energy bench- mark sets DFT-D4 slightly outperforms DFT-D3. Especially for metal containing systems, the introduced charge dependence improves thermochemical properties. We suggest (DFT-)D4 as a physically improved and more sophisticated dispersion model in place of DFT-D3 for DFT calculations as well as for other low-cost approaches like semi-empirical models. Keywords: density functional theory, London dispersion, non-covalent interactions, thermochemistry I. INTRODUCTION only mentioned briefly here. Among them are the ex- change hole dipole moment14,17{19 (XDM) model and the 20 Many computational studies have shown that dispersion- Tkatchenko-Scheffler (TS) model both employing Hir- corrected Kohn-Sham density functional theory (abbre- shfeld partitioning. It is also possible to directly develop viated as DFT in the following) is currently the method non-local density functionals, which are inherently capa- of choice for the routine computation of the electronic ble of describing London dispersion interactions. This and geometric structure of large systems1,2, e.g., in the way, atomic partitioning, which always involves some ar- fields of supra-molecular chemistry3{5, catalysis6,7, or bitrariness, is avoided. Here, the family of van der Waals in materials science8{10. In contrast to more elabo- density functionals is to be mentioned, which are based rate wave function theory (WFT) based methods, most on the fundamental adiabatic connection theory and of- density functional approximations (DFAs) are not able fer a rigorous basis for the design of dispersion-inclusive 21,22 to describe long-ranged electron correlation effects11{14. exchange-correlation functionals . A simplified con- Their treatment is, however, important to compute en- struction scheme for the non-local correlation part has 23,24 ergetic properties with high accuracy (less than about 1 been introduced by Vydrov and Van Voorhis . kcal mol−1 error), particularly for non-covalently bonded The computational costs of incorporating dispersion cor- or condensed phase systems. Therefore, various correc- rections into a standard DFT treatment are very method tion schemes have been developed to describe so-called dependent but generally smaller compared to the actual London dispersion interactions in a DFT framework15. DFT calculation. Consequently, it is reasonable to in- The most commonly used method for molecular appli- clude them by default, and it has been demonstrated cations is the so-called DFT-D3 scheme16, which cal- that the accuracy of dispersion-corrected DFAs for ther- culates the inter- and intramolecular dispersion interac- mochemical properties on average follows the Jacob's lad- tions only by employing the given system coordinates der25 classification. Note that for very low-cost atomistic (and atomic numbers). Similar atom pair-wise models, models like semi-empirical molecular orbital or force-field which additionally include information from the elec- methods, only non-density dependent schemes like DFT- tron density, have been reviewed recently15 and are D3 (and the here proposed D4) are computationally fea- sible. The current work describes the further development of a)Corresponding author:[email protected] the widely used semi-classical DFT-D3 approach. Re- 2 cently, it was shown that this \geometry-only" model xyz can be further improved by addition of atomic charge in- q formation26. Therein we showed that computed atom-in- tot molecule dynamic polarizabilities can be scaled by means of an element specific function with Mulliken-type atomic charges as input. A Σ AABqA+χB= 0 Here we want to report on the final version of the D4 A=1 model and provide it in a usable form for a large number (default) of density functionals as DFT-D4. In general we retained IN-/OUTPUT xyz Nel. the general idea and strong points of the well established Alternative D3 scheme and introduced the charge dependence as well STORED (optional) as some less important, mostly technical improvements. charges zA=ZA+qA Compared to the scaling scheme in Ref. 26, a less empir- AT RUNTIME (TB,DFT) ical function is used (see below) and the Mulliken partial charges are replaced by default with classical electroneg- αA,ref(iω) 1) Scaling ativity equilibration (EEQ) partial charges as recently A described by Goedecker et al27. The D4 scheme is, how- CN ever, general in a sense that any type of atomic charges zA,ref αA,ref(iω,zA) in addition to the geometric structure of the system can be used as input. CNA,ref 2) Averaging Previous studies have revealed that many-body disper- sion interactions beyond a pair-wise picture are impor- 28,29 30,31 tant, e.g., in supra-molecular , in cluster , and αA(iω,zA,CNA) in condensed-phase systems32{36 where so-called Dobson type B effects37 play an important role. The use of a coupled-dipole based many-body dispersion (MBD) cor- DFT-D4 Integration rection (cf. the MBD method in Refs. 38,39) in our imple- AB (optional) C6 mentation does not yield systematic improvements com- ABC pared to the simple third-order Axilrod-Teller-Muto40,41 A(iω)T C9 (ATM) term. However, the MBD correction proves to be Pair-wise Three-body Many-body useful for large systems (see non-covalent benchmarks in dispersion dispersion dispersion section III), for which we recommend its application in energy energy terms of an energy correction. The theory and technical details of the D4 method are described in section II. Subsequently, in section III, re- FIG. 1. Schematic workflow of the D4 program for an example sults for dispersion coefficients with D4 as well as for molecule (all definitions and steps are explained in detail in energies, and structures obtained with DFT-D4 are com- the text). pared directly to those of other established dispersion correction schemes for the same underlying DFA. Finally, a summary and an outlook on possible future work will atomic reference polarizabilities. For this purpose, a new be given. charge-scaling function ζ has been designed in this work given by ζ(zA; zA;ref ) = II. THEORY A;ref (2) A z exp β1 1 − exp γ 1 − Atomic units are used throughout in this work. The zA DFT-D4 workflow is simplified for an example molecule which is sketched exemplarily in Figure 2. In equation 2, in Figure 1. The pair-wise dipole-dipole dispersion coef- the chemical hardness γA is taken from Ref. 42 and serves ficients in D4 are calculated by numerical integration via as a non-fitted element-specific parameter to control the the well-known Casimir-Polder relation steepness of the scaling function. The value of the global 1 parameter β = 3 was determined by inspection. The 3 Z 1 AB A B empirical charge-dependent scaling in D4 is intended to C6 = d! α (i!)α (i!): (1) π increase the magnitude of the atomic dynamic dipole 0 polarizabilities α(i!) for larger number of electrons in At runtime, the isotropically averaged atomic dynamic proximity to the considered atom. Independent from the polarizabilities at imaginary frequency α(i!) in equa- actual DFA which is to be corrected, environment de- tion 1 are obtained in two steps: The first step incor- pendent atomic partial charges are taken as input. The porates an atomic partial charge dependent scaling of advantage is that such external partial charges can be 3 A 0 β1 and γ are set to unity TABLE I. Effective nuclear charges Z used in the D4 charge 2.5 scaling function for elements beyond krypton. The subtracted number corresponds to the number of core electrons absorbed in the ECP used (default def2-ECPs in TURBOMOLE43{45). 2 Element Z' ζ 1.5 Rb{Xe Z-18 Cs{La Z-46 Ce{Lu Z-28 1 Hf{Rn Z-60 0.5 0.5 1 1.5 2 2.5 3 3.5 lized. Since such atom-in-molecule dynamic polarizabili- zA,ref /zA ties cannot be calculated directly, molecular dynamic po- larizabilities of the reference systems AmXn (having m chemically equivalent atoms A and n chemically equiva- FIG.
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