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The adiabatic connection Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Dipartimento di Scienze Chimiche, Università degli Studi di Trieste, Trieste, Italy Department of Chemistry, University of Oslo, November 2 29 T. Helgaker (CTCC, University of Oslo) The adiabatic connection

Outline 1 The universal density functional and Lieb s theory 2 Evaluation of density functionals by Lieb maximization 3 The adiabatic connection 4 Dynamical correlation in the adiabatic connection 5 Static correlation in the adiabatic connection 6 Range-separated methods and the generalized adiabatic connection 7 Density- and potential-fixed adiabatic connections 8 Conclusions Overview

The Levy constrained-search functional The ground-state energy with external potential v and coupling strength λ: E λ [v] = inf Ψ N Ψ Hλ [v] Ψ H λ [v] = T + λw + P i v(r i ), W = P i>j r 1 ij, λ 1 It is possible to perform this minimization in two steps E λ [v] = inf F λ [ρ] + R v(r)ρ(r) dr Hohenberg Kohn variation principle ρ N F λ [ρ] = inf Ψ Hλ [] Ψ Levy constrained-search functional Ψ ρ In applications of DFT, approximations to F λ [ρ] are made typically assessed by comparison with experiment We are going to present rigorous evaluations of F λ [ρ] provide insight into F λ [ρ] and its dependence on static and dynamical correlation test approximate F λ [ρ] and suggest new ones Our tool will be the adiabatic connection the dependence of F λ [ρ] on λ for fixed ρ: F [ρ] F 1 [ρ] For such studies, a slightly different formulation of DFT is useful The universal density functional and Lieb s theory

The Lieb convex-conjugate functional In Lieb s theory, F λ [ρ] is defined as the convex conjugate to E λ [v]: F λ [ρ] = sup `Eλ [v] R v(r)ρ(r) dr the Lieb variation principle v E λ [v] = inf `Fλ [ρ] + R v(r)ρ(r) dr the Hohenberg Kohn variation principle ρ the two variation principles are Legendre Fenchel (LF) transforms The possibility of the LF formulation follows from the convexity of E λ [v] in v: f x1 cf x1 1 c f x2 f c x1 1 c x2 f x2 x1 c x1 1 c x2 x2 it then has a convex conjugate partner: the Lieb functional F λ [ρ] conjugate functions have inverse first derivatives A convex functional and its conjugate partner satisfy Fenchel s inequality: F λ [ρ] E λ [v] R v(r)ρ(r) dr E λ [v] F λ [ρ] + R v(r)ρ(r) dr either variation principle sharpens Fenchel s inequality into an equality We shall use Lieb s variation principle to calculate F λ [ρ] at different levels of theory The universal density functional and Lieb s theory

Lieb s theory for approximate energies Lieb s theory may be applied directly to any approximate energy that is concave: Fλ mod [ρ] LF Eλ mod [v] examples: all variationally determined energies such as Eλ HF [v] and E FCI λ [v] For nonconcave energies, Fλ mod [ρ] is well-defined but not strictly conjugate to Eλ mod[v] instead, Fλ mod [ρ] is conjugate to the least concave upper bound to the energy Eλ mod [v] LF Fλ mod [ρ] LF e E mod λ [v] E mod [v] λ 4 4 3 3 2 2 1 1 4 2 2 4 1 2 4 2 2 4 1 2 examples: E MP2 λ [v] and Eλ CCSD [v] Approximate energies

Lieb maximization For a given density ρ(r) and chosen level of theory E λ [v], we wish to calculate Z «F λ [ρ] = max E λ [v] v(r)ρ(r) dr v by maximizing the right-hand side with respect to variations in the potential v(r) The potential is parameterized as v c(r) = v ext(r) + (1 λ)v ref (r) + X t c t g t(r) where the three terms are the physical, external potential v ext(r) the Fermi Amaldi reference potential to ensure correct asymptotic behaviour v ref (r) = 1 1 «Z ρ(r ) N r r dr an expansion in Gaussians g t(r) with coefficients c t we use large orbital basis sets, typically augmented with diffuse functions Implemented levels of theory of E λ [v] are: HF, MP2, CCD, CCSD, CCSD(T), FCI Wu and Yang (23), Teale, Coriani and Helgaker (29) Lieb maximization

First- and second-order Lieb maximizations We determine v c(r) and hence F λ [ρ] by maximizing with respect to c t the quantity Z G λ,ρ (c) = E λ [v c] v c(r)ρ(r) dr v c(r) = v ext(r) + (1 λ)v ref (r) + P t ct gt(r) Our convergence target is a gradient norm smaller than 1 6 The quasi-newton method requires only the gradient Z G λ,ρ (c) = ˆρλ,c(r) ρ(r) g t(r) dr c t implemented with BFGS update converges in 1 2 iterations The Newton method requires also the Hessian 2 Z Z G λ,ρ (c) = g t(r)g u(r ) δρ(r) c t c u δv(r dr dr ) calculated (exactly or approximately) from (CCSD) linear response theory expensive but robust convergence converges in 5 1 iterations All code is implemented in DALTON Lieb maximization

The adiabatic connection HF, MP2, CCSD and CCSD(T) plots of F λ [ρ] for the neon atom in the cc-pvqz basis 18 17 FΛ Ρ a.u. 16 15 14 13 Λ These are very boring curves: almost indistinguishable and nearly linear F λ [ρ] = min Ψ T + λw Ψ slight concavity from variation principle Ψ ρ Note the noninteracting limit: Ψ F [ρ] = min Ψ T = Ts[ρ] Ψ ρ noninteracting kinetic energy Lieb maximization

The exchange correlation functional We now subtract off the boring parts: the noninteracting limit and the Coulomb energy: E xc,λ [ρ] = F λ [ρ] T s[ρ] λj[ρ] exchange correlation energy ZZ J[ρ] = ρ(r 1 )ρ(r 2 )r 1 12 dr 1dr 2 Coulomb energy The resulting curves are negative but still concave since a linear term has been subtracted: 2 4 EXC, Λ Ρ a.u. 6 8 1 12 Λ the exchange energy dominates these curves (this is not always true) we can now distinguish the HF curve (top) from the remaining ones Lieb maximization

The exchange and correlation functionals The exchange correlation functional is typically dominated by the exchange functional: E x,λ [ρ] = λ ` Ψ Ψ W J[ρ] < exchange functional the exchange energy depends linearly on λ Subtracting the exchange functional, we obtain the correlation functional: E c,λ [ρ] = E xc,λ [ρ] E x,λ [ρ] < correlation functional The resulting AC correlation curve is negative and concave..5.1 EC,Λ Ρ a.u..15.2.25.3.35 Λ the HF curve is (nearly) horizontal and (nearly) zero the MP2, CCSD and CCSD(T) curves have are (nearly) quadratic shape Lieb maximization

The AC integrand Being concave, E xc,λ [ρ] may be represented in terms of a decreasing AC integrand: Z λ Z λ E xc,λ [ρ] = E xc,λ [ρ] dλ = W xc,λ [ρ] dλ (W xc,λ [ρ] decreasing) The Hellmann Feynman theorem provides an explicit expression for the AC integrand: W xc,λ [ρ] = Ψ λ W Ψλ J[ρ] (AC integrand) 2 2 4 4 6 8 E xc, Λ Ρ 6 8 area E xc, Λ Ρ 1 1 12 12 14 W xc, Λ Ρ note: the AC integrand has a large constant exchange contribution (above the horizontal line) Langreth and Perdew 1975, Gunnarsson and Lindqvist 1976 The adiabatic connection

A geometrical view of the adiabatic connection The exchange energy of H 2 at R = 5a E X W Ψ W Λ ' Ψ J Λ ' Ρ E X WXC,Λ a.u...2.4.6.8 The adiabatic connection

A geometrical view of the adiabatic connection The (total) correlation energy of H 2 at R = 5a E X W Ψ W Λ ' Ψ J Λ ' Ρ E X WXC,Λ a.u. E C T C W 1 Ψ 1 W Λ ' Ψ 1 J Λ ' Ρ..2.4.6.8 The adiabatic connection

A geometrical view of the adiabatic connection The kinetic correlation energy of H 2 at R = 5a E X W Ψ W Λ ' Ψ J Λ ' Ρ E X WXC,Λ a.u. E C T C W 1 Ψ 1 W Λ ' Ψ 1 J Λ ' Ρ..2.4.6.8 The adiabatic connection

A geometrical view of the adiabatic connection The exchange correlation energy of H 2 at R = 5a E X W Ψ W Λ ' Ψ J Λ ' Ρ E X WXC,Λ a.u. E C T C W 1 Ψ 1 W Λ ' Ψ 1 J Λ ' Ρ..2.4.6.8 The adiabatic connection

Review and preview The energy may equivalently be expressed as a functional of density or potential E λ [v] = inf ρ `Fλ [ρ] + R v(r)ρ(r) dr the Hohenberg Kohn variation principle F λ [ρ] = sup v `Eλ [v] R v(r)ρ(r) dr the Lieb variation principle we normally generate E λ [v] by minimization with respect to ρ(r) here we consider the opposite, generating F λ [ρ] by maximization with respect v(r) This conjugate relationship holds at all levels for which E λ [v] is concave E λ [v] LF F λ [ρ] LF con e Eλ [v] E λ [v] We have discussed the general shape of AC curves Coulomb, exchange and correlation We shall now calculate F λ [ρ] at different levels of theory different basis sets and basis-set convergence neon and water (dynamic correlation) dissociation of H 2 (static correlation) We shall finally consider generalized adiabatic connections spatial decomposition of the exchange correlation energy Review and preview

Basis-set convergence of the AC integrand He atom in the aug-cc-pvdz basis Directly calculated and AC-integrated FCI correlation energies agree to within 9 µh 1.2 1.4 E XC aug cc pvdz 1.4964 a.u. E XC aug cc pvdz FCI 1.4973 a.u..9 a.u. WΛ 1.6 1.8 1.1..2.4.6.8 Review and preview

Basis-set convergence of the AC integrand He atom in the aug-cc-pvtz basis Agreement is improved by an order of magnitude in the larger TZ basis to 1 µh 1.2 1.4 E XC aug cc pvtz 1.6252 a.u. E XC aug cc pvtz FCI 1.6253 a.u..1 a.u. WΛ 1.6 1.8 1.1..2.4.6.8 Review and preview

Basis-set convergence of the AC integrand He atom in the aug-cc-pvqz basis In the QZ basis, the energies agree to all quoted digits 1.2 E XC aug cc pvqz 1.65 a.u. E XC aug cc pvqz FCI 1.65 a.u. 1.4 WΛ 1.6 1.8 1.1..2.4.6.8 Review and preview

Basis-set convergence of the AC integrand He atom in the aug-cc-pv5z basis Basis-set convergence is the usual, slow X 3 convergence 1.2 E XC aug cc pv5z 1.6597 a.u. E XC aug cc pv5z FCI 1.6597 a.u. 1.4 WΛ 1.6 1.8 1.1..2.4.6.8 Review and preview

Basis-set convergence of the AC integrand He atom in the aug-cc-pv6z basis In the largest aug-cc-pv6z basis, exchange correlation energy is 1.663 mh 1.2 E XC aug cc pv6z 1.6631 a.u. E XC aug cc pv6z FCI 1.6631 a.u. 1.4 WΛ 1.6 1.8 1.1..2.4.6.8 Review and preview

Basis-set convergence of the AC integrand Extrapolated FCI basis-set limit of the He atom Two-point basis-set extrapolation yields an exchange correlation energy of 1.668 mh 1.2 E XC Extrap. 1.6677 a.u. E XC Extrap FCI 1.6678 a.u. 1.4 WΛ 1.6 1.8 1.1..2.4.6.8 Review and preview

The Hartree Fock correlation energy... the neon atom The HF energy has a tiny correlation contribution ( 1.4 mh for neon) the HF state and its KS reference state are different the associated orbital-relaxation energy is proportional to λ 2 12.175 12.18 E c Ρ WXC,Λ a.u. 12.185 12.19 T c Ρ 12.195 12.11 Λ The AC correlation curve is therefore very nearly a straight line the kinetic and total energies follow the virial theorem In HF-OEP theory, the energy is slightly higher than the HF energy (by 1.7 mh) in OEP, we insist on a local potential for an explicit exchange functional we likewise have a local potential but have no constraints on the functional Review and preview

The Hartree Fock correlation energy... the water molecule The HF energy has a tiny correlation contribution ( 2.1 mh for water) the HF state and its KS reference state are different the associated orbital-relaxation energy is proportional to λ 2 8.956 E c Ρ WXC,Λ a.u. 8.957 8.958 T c Ρ 8.959 Λ The AC correlation curve is therefore very nearly a straight line the kinetic and total energies follow the virial theorem In HF-OEP theory, the energy is slightly higher than the HF energy (by 2.2 mh) in OEP, we insist on a local potential for an explicit exchange functional we likewise have a local potential but have no constraints on the functional Review and preview

The MP2 and CCSD correlation energies the neon atom The MP2 and CCSD curves are not straight but bend slightly upwards the MP2 curve lies below the CCSD curve 12.1 12.2 E c Ρ WXC,Λ a.u. 12.3 12.4 12.5 12.6 T c Ρ 12.7 Λ A linear curve would reflect a λ 2 dependence of the correlation energy why does the curve bend? Review and preview

The MP2 and CCSD correlation energies the water molecule The MP2 and CCSD curves are not straight but bend slightly upwards the MP2 curve lies below the CCSD curve 9. 9.1 E c Ρ WXC,Λ a.u. 9.2 9.3 T c Ρ 9.4 9.5 Λ A linear curve would reflect a λ 2 dependence of the correlation energy why does the curve bend? Review and preview

Two-parameter CCSD correlation curves Second-order Görling Levy theory suggests the following model: E c,λ [ρ] = λ2 g 2 h λg = sλ2 /2 1 + 2aλ W c,λ [ρ] = sλ + saλ2 (1 + 2aλ), a = 2s 2t 2 s 2 + 8st s + 4t here s is the initial slope, while t is the end-point value of the curve This two-parameter model agrees perfectly with the calculated AC points: the denominator opens up the HOMO LUMO gap and is responsible for convexity..1.2 CCSD water WΛ a.u..3.4 CCSD neon.5.6 Λ Review and preview

The adiabatic connection: the triples correction the neon atom The CCSD(T) curve lies slightly below the CCSD curve the difference increases with increasing λ 12.1. 12.2 E c Ρ.5 12.3 WXC,Λ a.u. 12.4 12.5 12.6 T c Ρ CCSD.1.15 W CCSD T Ρ W CCSD Ρ 12.7 CCSD T An inspection of the triples correction suggests that it behaves like λ 3 the difference between the CCSD and CCSD(T) integrands should behave like λ 2 an almost perfect fit is possible with a one-parameter quadratic curve W CCSD(T) c,λ [ρ] Wc,λ CCSD [ρ].189λ2 Review and preview

The adiabatic connection: the triples correction the water molecule The CCSD(T) curve lies slightly below the CCSD curve the difference increases with increasing λ. 9. 9.1 E c Ρ.5 WXC,Λ a.u. 9.2 9.3.1.15 W CCSD T Ρ W CCSD Ρ 9.4 T c Ρ CCSD.2 9.5 CCSD T.25 Λ An inspection of the triples correction suggests that it behaves like λ 3 the difference between the CCSD and CCSD(T) integrands should behave like λ 2 an almost perfect fit is possible with a one-parameter quadratic curve W CCSD(T) c,λ [ρ] Wc,λ CCSD [ρ].256λ2 Review and preview

The construction of DFT functionals from AC models Part of the motivation for studying the AC is to develop new functionals simple forms can be turned directly into functionals by integration famous example: the Becke HH functional by linear interpolation Note: all such AC forms are only as good as their input data we must also provide a recipe for producing input data Previous work: Ernzerhof (1996) Burke et al. (1997) Seidl et al. (2) Mori-Sánchez et al. (27) Peach et al. (28) In the absence of accurate data, there was little to differentiate the various AC forms none of the suggested forms reproduce our data We have studied the AC curves for systems dominated by dynamical correlation we shall now see if they are sufficiently flexible for static correlation Review and preview

The adiabatic connection: the dissociation of H 2 To study static correlation, we consider H 2 dissociation RHF, BLYP, and FCI levels of theory in the aug-cc-pvqz basis.5.6.7.8 HF.9 1. BLYP FCI 1.1 1.2 Dissociation of H 2 and static correlation

Adiabatic connection: XC curves for H 2 R =.7 bohr At short distances, exchange (.827E h ) dominates over correlation (.39E h ) energy BLYP curve performs well, reproducing HF exchange and FCI correlation The BLYP and CI curves are nearly linear, indicative of dynamical correlation.4.5.6.7.8 HF BLYP.9 R.7 bohr FCI Dissociation of H 2 and static correlation

Adiabatic connection: XC curves for H 2 R = 1.4 bohr The overall picture is similar to that at R =.7 bohr Most notably, the exchange energy increases from 827 to 661 me h The correlation energy decreases slightly, from 39 to 41 me h.4.5.6.7.8 HF BLYP FCI.9 R 1.4 bohr Dissociation of H 2 and static correlation

Adiabatic connection: XC curves for H 2 R = 3. bohr The exchange energy increases further from 661 to 477 me h The correlation energy decreases from 41 to 77 me h The FCI curve now curves more strongly, indicative of static correlation The BLYP functional overestimates exchange but works well by LYP error cancellation.4.5.6 HF BLYP FCI.7.8.9 R 3. bohr Dissociation of H 2 and static correlation

Adiabatic connection: XC curves for H 2 R = 5. bohr The overall picture is similar to that at R = 3. bohr However, the static-correlation curvature of the FCI curve is now more pronounced The (nonconvex) BLYP curve strongly underestimates the magnitude of the XC energy The BLYP curve now benefits less from error cancellation.4 HF.5.6 BLYP FCI.7.8.9 R 5. bohr Dissociation of H 2 and static correlation

Adiabatic connection: XC curves for H 2 R = 1. bohr At R = 1 bohr, the two atoms are separated and correlation is essentially static Dynamical correlation is now less than 1 E h (dispersion) The BLYP functional works mostly by overestimating the exchange energy It still has a nonzero slope, indicative of dynamic correlation!.4 HF.5.6 BLYP FCI.7.8.9 R 1. bohr Dissociation of H 2 and static correlation

Adiabatic connection: correlation-only curves for H 2 R =.7 bohr The correlation-only AC curve illustrates the performance of the LYP functional it performs in much the same manner for helium and two separated hydrogen atoms its construction was based on the helium atom Below, blue and yellow areas represent kinetic and total correlation FCI energies. HF.5.1 FCI LYP.15.2 R.7 bohr.25 Dissociation of H 2 and static correlation

Adiabatic connection: correlation-only curves for H 2 R = 1.4 bohr The correlation-only AC curve illustrates the performance of the LYP functional it performs in much the same manner for helium and two separated hydrogen atoms its construction was based on the helium atom Below, blue and yellow areas represent kinetic and total correlation FCI energies. HF.5.1 FCI LYP.15.2 R.7 bohr.25 Dissociation of H 2 and static correlation

Adiabatic connection: correlation-only curves for H 2 R = 3. bohr The correlation-only AC curve illustrates the performance of the LYP functional it performs in much the same manner for helium and two separated hydrogen atoms its construction was based on the helium atom Below, blue and yellow areas represent kinetic and total correlation FCI energies. HF.5.1 LYP FCI.15.2 R 3. bohr.25 Dissociation of H 2 and static correlation

Adiabatic connection: correlation-only curves for H 2 R = 5. bohr The correlation-only AC curve illustrates the performance of the LYP functional it performs in much the same manner for helium and two separated hydrogen atoms its construction was based on the helium atom Below, blue and yellow areas represent kinetic and total correlation FCI energies. HF.5 LYP.1.15.2.25 R 5. bohr FCI Dissociation of H 2 and static correlation

Adiabatic connection: correlation-only curves for H 2 R = 7. bohr The correlation-only AC curve illustrates the performance of the LYP functional it performs in much the same manner for helium and two separated hydrogen atoms its construction was based on the helium atom Below, blue and yellow areas represent kinetic and total correlation FCI energies. HF.5 LYP.1.15.2.25 R 7. bohr FCI Dissociation of H 2 and static correlation

Adiabatic connection: XC curves for H 2 R = 1. bohr The correlation-only AC curve illustrates the performance of the LYP functional it performs in much the same manner for helium and two separated hydrogen atoms its construction was based on the helium atom Below, blue and yellow areas represent kinetic and total correlation FCI energies. HF.5 LYP.1.15.2 R 1. bohr.25 FCI Dissociation of H 2 and static correlation

The two-parameter model applied to the H 2 molecule Application of the two-parameter model to the FCI dissociation of H 2 : f (λ, s, t) = sλ + saλ2 (1 + 2aλ) 2, a = 2s 2t s 2 + 8st s + 4t the dashed lines have been obtained by using the exact initial gradient and exact end point the full lines have been obtained by a least-squares fit to all points..5.1.15.2.25 Dissociation of H 2 and static correlation

Generalized, range-separated adiabatic connection Up to now, we have studied the AC with a uniformly scaled two-electron interaction We shall now consider range-separated adiabatic interactions w s λ (r ij ) = λ r ij λ erf w e λ (r 1 λ r ij ij ) = r ij λ erf w g λ (r 1 λ r ij ij ) = 2 λ r ij π 1 λ «exp 1 λ 3 1 λ «2 «r 2 ij 8 4 standard 8 4 error function damped 8 4 Gaussian attenuated error function damped As λ increases, we build up the full interaction from outer to inner region this provides an alternative view of correlation such ACs are of particular relevance for range-separated methods Savin et al. 1996, Yang 1998, Toulouse et al. 24 Generalized adiabatic connections

6 5 4 3 2 1 6 5 4 3 2 1 Damping with error function λ =. Comparison of the full Coulomb potential with the err and err gau potentials λ erf w e λ (r 1 λ r ij ij ) = r ij λ erf w g λ (r 1 λ r ij ij ) = 2 λ r ij π 1 λ «exp 1 λ 3 1 λ «2 «r 2 ij err potential err gau potential erf erf gau Generalized adiabatic connections

6 5 4 3 2 1 6 5 4 3 2 1 Damping with error function λ =.1 Comparison of the full Coulomb potential with the err and err gau potentials λ erf w e λ (r 1 λ r ij ij ) = r ij λ erf w g λ (r 1 λ r ij ij ) = 2 λ r ij π 1 λ «exp 1 λ 3 1 λ «2 «r 2 ij err potential err gau potential erf erf gau Generalized adiabatic connections

6 5 4 3 2 1 6 5 4 3 2 1 Damping with error function λ =.2 Comparison of the full Coulomb potential with the err and err gau potentials λ erf w e λ (r 1 λ r ij ij ) = r ij λ erf w g λ (r 1 λ r ij ij ) = 2 λ r ij π 1 λ «exp 1 λ 3 1 λ «2 «r 2 ij err potential err gau potential erf erf gau Generalized adiabatic connections

6 5 4 3 2 1 6 5 4 3 2 1 Damping with error function λ =.3 Comparison of the full Coulomb potential with the err and err gau potentials λ erf w e λ (r 1 λ r ij ij ) = r ij λ erf w g λ (r 1 λ r ij ij ) = 2 λ r ij π 1 λ «exp 1 λ 3 1 λ «2 «r 2 ij err potential err gau potential erf erf gau Generalized adiabatic connections

6 5 4 3 2 1 6 5 4 3 2 1 Damping with error function λ =.4 Comparison of the full Coulomb potential with the err and err gau potentials λ erf w e λ (r 1 λ r ij ij ) = r ij λ erf w g λ (r 1 λ r ij ij ) = 2 λ r ij π 1 λ «exp 1 λ 3 1 λ «2 «r 2 ij err potential err gau potential erf erf gau Generalized adiabatic connections

6 5 4 3 2 1 6 5 4 3 2 1 Damping with error function λ =.5 Comparison of the full Coulomb potential with the err and err gau potentials λ erf w e λ (r 1 λ r ij ij ) = r ij λ erf w g λ (r 1 λ r ij ij ) = 2 λ r ij π 1 λ «exp 1 λ 3 1 λ «2 «r 2 ij err potential err gau potential erf erf gau Generalized adiabatic connections

6 5 4 3 2 1 6 5 4 3 2 1 Damping with error function λ =.6 Comparison of the full Coulomb potential with the err and err gau potentials λ erf w e λ (r 1 λ r ij ij ) = r ij λ erf w g λ (r 1 λ r ij ij ) = 2 λ r ij π 1 λ «exp 1 λ 3 1 λ «2 «r 2 ij err potential err gau potential erf erf gau Generalized adiabatic connections

6 5 4 3 2 1 6 5 4 3 2 1 Damping with error function λ =.7 Comparison of the full Coulomb potential with the err and err gau potentials λ erf w e λ (r 1 λ r ij ij ) = r ij λ erf w g λ (r 1 λ r ij ij ) = 2 λ r ij π 1 λ «exp 1 λ 3 1 λ «2 «r 2 ij err potential err gau potential erf erf gau Generalized adiabatic connections

6 5 4 3 2 1 6 5 4 3 2 1 Damping with error function λ =.8 Comparison of the full Coulomb potential with the err and err gau potentials λ erf w e λ (r 1 λ r ij ij ) = r ij λ erf w g λ (r 1 λ r ij ij ) = 2 λ r ij π 1 λ «exp 1 λ 3 1 λ «2 «r 2 ij err potential err gau potential erf erf gau Generalized adiabatic connections

6 5 4 3 2 1 6 5 4 3 2 1 Damping with error function λ =.9 Comparison of the full Coulomb potential with the err and err gau potentials λ erf w e λ (r 1 λ r ij ij ) = r ij λ erf w g λ (r 1 λ r ij ij ) = 2 λ r ij π 1 λ «exp 1 λ 3 1 λ «2 «r 2 ij err potential err gau potential erf erf gau Generalized adiabatic connections

6 5 4 3 2 1 6 5 4 3 2 1 Damping with error function λ = 1. Comparison of the full Coulomb potential with the err and err gau potentials λ erf w e λ (r 1 λ r ij ij ) = r ij λ erf w g λ (r 1 λ r ij ij ) = 2 λ r ij π 1 λ «exp 1 λ 3 1 λ «2 «r 2 ij err potential err gau potential erf erf gau Generalized adiabatic connections

AC correlation curves for the He isoelectronic series Z = 1 Standard AC curve on the left linearity increases with increasing Z Range-separated (erf gau) AC curve on the right curves reveal increasing compactness with increasing Z...1.2.2.4 WΛ,c a.u..3.4.6.8.5.1.6.12.7.14 Generalized adiabatic connections

AC correlation curves for the He isoelectronic series Z = 2 Standard AC curve on the left linearity increases with increasing Z Range-separated (erf gau) AC curve on the right curves reveal increasing compactness with increasing Z...5.2 WΛ,c a.u..4.1.15.6.2.25.8 Generalized adiabatic connections

AC correlation curves for the He isoelectronic series Z = 3 Standard AC curve on the left linearity increases with increasing Z Range-separated (erf gau) AC curve on the right curves reveal increasing compactness with increasing Z...1.2.1 WΛ,c a.u..3.4 WΛ,c a.u..2.5.6.3.7.4 Λ Generalized adiabatic connections

AC correlation curves for the He isoelectronic series Z = 4 Standard AC curve on the left linearity increases with increasing Z Range-separated (erf gau) AC curve on the right curves reveal increasing compactness with increasing Z...1.2.2 WΛ,c a.u..4 WΛ,c a.u..3.6.4.5 Λ Generalized adiabatic connections

AC correlation curves for the He isoelectronic series Z = 5 Standard AC curve on the left linearity increases with increasing Z Range-separated (erf gau) AC curve on the right curves reveal increasing compactness with increasing Z...1.2.2 WΛ,c a.u..4 WΛ,c a.u..3.4.6.5.6 Λ Generalized adiabatic connections

AC correlation curves for the He isoelectronic series Z = 6 Standard AC curve on the left linearity increases with increasing Z Range-separated (erf gau) AC curve on the right curves reveal increasing compactness with increasing Z...2.2 WΛ,c a.u..4 WΛ,c a.u..4.6.6.8 Λ Generalized adiabatic connections

AC correlation curves for the He isoelectronic series Z = 7 Standard AC curve on the left linearity increases with increasing Z Range-separated (erf gau) AC curve on the right curves reveal increasing compactness with increasing Z...2.2 WΛ,c a.u..4 WΛ,c a.u..4.6.6.8.8 Λ Generalized adiabatic connections

AC correlation curves for the He isoelectronic series Z = 8 Standard AC curve on the left linearity increases with increasing Z Range-separated (erf gau) AC curve on the right curves reveal increasing compactness with increasing Z...2.2.4 WΛ,c a.u..4 WΛ,c a.u..6.6.8 1..8 Λ Generalized adiabatic connections

AC correlation curves for the He isoelectronic series Z = 9 Standard AC curve on the left linearity increases with increasing Z Range-separated (erf gau) AC curve on the right curves reveal increasing compactness with increasing Z...2.2.4 WΛ,c a.u..4 WΛ,c a.u..6.8.6 1..8 1.2 Λ Generalized adiabatic connections

AC correlation curves for the He isoelectronic series Z = 1 Standard AC curve on the left linearity increases with increasing Z Range-separated (erf gau) AC curve on the right curves reveal increasing compactness with increasing Z...2.2.4 WΛ,c a.u..4 WΛ,c a.u..6.8.6 1. 1.2.8 1.4 Λ Generalized adiabatic connections

Range-separated AC correlation curves for Be and Ne Correlation-only (erf) curves for Be (left) and Ne (right) MP2 (yellow), CCSD (red) and CCSD(T) (blue) The Be atom is more diffuse than the Ne atom The MP2 model underestimates long-range interactions..2.2 WC,Λ a.u..4 WC,Λ a.u..4.6.6.8.8 Λ Λ Generalized adiabatic connections

Range separation: dissociation of H 2 the total AC curve at internuclear separation.7 bohr We consider the total AC curve first includes Coulomb, exchange and correlation contributions Standard AC curve on the left it undergoes the usual transition from a sloped to horizontal curve at full separation this reflects the transition from dynamical to static correlation Range-separated erf gau AC curve on the right it moves towards small λ values with increasing separation at full separation, all total interactions are interatomic 2. 2. 1.5 1.5 WΛ a.u. 1. WΛ a.u. 1..5.5...2.4.6.8 Λ. Λ Generalized adiabatic connections

Range separation: dissociation of H 2 the total AC curve at internuclear separation 1.4 bohr We consider the total AC curve first includes Coulomb, exchange and correlation contributions Standard AC curve on the left it undergoes the usual transition from a sloped to horizontal curve at full separation this reflects the transition from dynamical to static correlation Range-separated erf gau AC curve on the right it moves towards small λ values with increasing separation at full separation, all total interactions are interatomic 2. 2. 1.5 1.5 WΛ a.u. 1. WΛ a.u. 1..5.5...2.4.6.8 Λ. Λ Generalized adiabatic connections

Range separation: dissociation of H 2 the total AC curve at internuclear separation 3. bohr We consider the total AC curve first includes Coulomb, exchange and correlation contributions Standard AC curve on the left it undergoes the usual transition from a sloped to horizontal curve at full separation this reflects the transition from dynamical to static correlation Range-separated erf gau AC curve on the right it moves towards small λ values with increasing separation at full separation, all total interactions are interatomic 2. 2. 1.5 1.5 WΛ a.u. 1. WΛ a.u. 1..5.5...2.4.6.8 Λ. Λ Generalized adiabatic connections

Range separation: dissociation of H 2 the total AC curve at internuclear separation 5. bohr We consider the total AC curve first includes Coulomb, exchange and correlation contributions Standard AC curve on the left it undergoes the usual transition from a sloped to horizontal curve at full separation this reflects the transition from dynamical to static correlation Range-separated erf gau AC curve on the right it moves towards small λ values with increasing separation at full separation, all total interactions are interatomic 2. 2. 1.5 1.5 WΛ a.u. 1. WΛ a.u. 1..5.5...2.4.6.8 Λ. Λ Generalized adiabatic connections

Range separation: dissociation of H 2 the total AC curve at internuclear separation 7. bohr We consider the total AC curve first includes Coulomb, exchange and correlation contributions Standard AC curve on the left it undergoes the usual transition from a sloped to horizontal curve at full separation this reflects the transition from dynamical to static correlation Range-separated erf gau AC curve on the right it moves towards small λ values with increasing separation at full separation, all total interactions are interatomic 2. 2. 1.5 1.5 WΛ a.u. 1. WΛ a.u. 1..5.5...2.4.6.8 Λ. Λ Generalized adiabatic connections

Range separation: dissociation of H 2 the total AC curve at internuclear separation 1. bohr We consider the total AC curve first includes Coulomb, exchange and correlation contributions Standard AC curve on the left it undergoes the usual transition from a sloped to horizontal curve at full separation this reflects the transition from dynamical to static correlation Range-separated erf gau AC curve on the right it moves towards small λ values with increasing separation at full separation, all total interactions are interatomic 2. 2. 1.5 1.5 WΛ a.u. 1. WΛ a.u. 1..5.5...2.4.6.8 Λ...2.4.6.8 Λ Generalized adiabatic connections

Range separation: dissociation of H 2 the correlation-only AC curve at internuclear separation.7 bohr We next consider the correlation-only AC curve Standard AC curve on the left it undergoes the same transition from a sloped to horizontal curve this reflects the transition from dynamical to static correlation Range-separated erf gau AC curve on the right at short bond distance, the interactions are predominantly short-ranged at long distances, short- and long-ranged interactions partially cancel.4.4.2.2.. WΛ,c a.u..2 WΛ,c a.u..2.4.4.6.6..2.4.6.8 Λ Static correlation is not easily treated by range separation Generalized adiabatic connections

Range separation: dissociation of H 2 the correlation-only AC curve at internuclear separation 1.4 bohr We next consider the correlation-only AC curve Standard AC curve on the left it undergoes the same transition from a sloped to horizontal curve this reflects the transition from dynamical to static correlation Range-separated erf gau AC curve on the right at short bond distance, the interactions are predominantly short-ranged at long distances, short- and long-ranged interactions partially cancel.4.4.2.2.. WΛ,c a.u..2 WΛ,c a.u..2.4.4.6.6..2.4.6.8 Λ Static correlation is not easily treated by range separation Generalized adiabatic connections

Range separation: dissociation of H 2 the correlation-only AC curve at internuclear separation 3. bohr We next consider the correlation-only AC curve Standard AC curve on the left it undergoes the same transition from a sloped to horizontal curve this reflects the transition from dynamical to static correlation Range-separated erf gau AC curve on the right at short bond distance, the interactions are predominantly short-ranged at long distances, short- and long-ranged interactions partially cancel.4.4.2.2.. WΛ,c a.u..2 WΛ,c a.u..2.4.4.6.6..2.4.6.8 Λ Static correlation is not easily treated by range separation Generalized adiabatic connections

Range separation: dissociation of H 2 the correlation-only AC curve at internuclear separation 5. bohr We next consider the correlation-only AC curve Standard AC curve on the left it undergoes the same transition from a sloped to horizontal curve this reflects the transition from dynamical to static correlation Range-separated erf gau AC curve on the right at short bond distance, the interactions are predominantly short-ranged at long distances, short- and long-ranged interactions partially cancel.4.4.2.2.. WΛ,c a.u..2 WΛ,c a.u..2.4.4.6.6..2.4.6.8 Λ Static correlation is not easily treated by range separation Generalized adiabatic connections

Range separation: dissociation of H 2 the correlation-only AC curve at internuclear separation 7. bohr We next consider the correlation-only AC curve Standard AC curve on the left it undergoes the same transition from a sloped to horizontal curve this reflects the transition from dynamical to static correlation Range-separated erf gau AC curve on the right at short bond distance, the interactions are predominantly short-ranged at long distances, short- and long-ranged interactions partially cancel.4.4.2.2.. WΛ,c a.u..2 WΛ,c a.u..2.4.4.6.6..2.4.6.8 Λ Static correlation is not easily treated by range separation Generalized adiabatic connections

Range separation: dissociation of H 2 the correlation-only AC curve at internuclear separation 1. bohr We next consider the correlation-only AC curve Standard AC curve on the left it undergoes the same transition from a sloped to horizontal curve this reflects the transition from dynamical to static correlation Range-separated erf gau AC curve on the right at short bond distance, the interactions are predominantly short-ranged at long distances, short- and long-ranged interactions partially cancel.4.4.2.2.. WΛ,c a.u..2 WΛ,c a.u..2.4.4.6.6..2.4.6.8 Λ Static correlation is not easily treated by range separation Generalized adiabatic connections

Density-fixed and potential-fixed adiabatic connections In Lieb s theory, E λ [v] and F λ [ρ] are conjugate functionals F λ [ρ] LF E λ [v] The two quantities may be manipulated in the same manner in particular, the AC connection exist for both functionals F λ [ρ] = F [ρ] + R λ F λ [ρ] dλ density-fixed AC (Görling Levy perturbation theory) E λ [v] = E [v] + R λ E λ [v] dλ potential-fixed AC (bare-nucleus perturbation theory) Below, we have plotted the total AC curves for the dissociation of the H 2 molecule potential-fixed AC (dashed lines) and density-fixed AC (full lines) 1..8 WΛ a.u..6.4.2 Λ Generalized adiabatic connections

Conclusions We have calculated the universal density functional by Lieb maximization Newton and quasi-newton methods many standard levels of theory: HF, MP2, CCSD, CCSD(T), BLYP many-electron atoms and molecules We have presented accurate adiabatic-connection (AC) curves total, exchange correlation and correlation-only curves We have discussed the shape of the AC curves static correlation give horizontal curves dynamical correlation gives slanted curves doubles give approximately linear curves; triples add quadratic correction these curves can be accurately modeled with few parameters We have studied the dissociation of H 2 HF, BLYP and FCI We have presented range-separated AC curves short- and long-range correlation static and dynamical correlation Conclusions