The adiabatic connection

Size: px
Start display at page:

Download "The adiabatic connection"

Transcription

1 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

2 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

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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

12 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 examples: E MP2 λ [v] and Eλ CCSD [v] Approximate energies

13 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

14 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

15 The adiabatic connection HF, MP2, CCSD and CCSD(T) plots of F λ [ρ] for the neon atom in the cc-pvqz basis FΛ Ρ a.u 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

16 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 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

17 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 Λ the HF curve is (nearly) horizontal and (nearly) zero the MP2, CCSD and CCSD(T) curves have are (nearly) quadratic shape Lieb maximization

18 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) E xc, Λ Ρ 6 8 area E xc, Λ Ρ 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

19 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 The adiabatic connection

20 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 The adiabatic connection

21 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 The adiabatic connection

22 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 The adiabatic connection

23 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

24 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 E XC aug cc pvdz a.u. E XC aug cc pvdz FCI a.u..9 a.u. WΛ Review and preview

25 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 E XC aug cc pvtz a.u. E XC aug cc pvtz FCI a.u..1 a.u. WΛ Review and preview

26 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 Review and preview

27 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 a.u. E XC aug cc pv5z FCI a.u. 1.4 WΛ Review and preview

28 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 mh 1.2 E XC aug cc pv6z a.u. E XC aug cc pv6z FCI a.u. 1.4 WΛ Review and preview

29 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 mh 1.2 E XC Extrap a.u. E XC Extrap FCI a.u. 1.4 WΛ Review and preview

30 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 E c Ρ WXC,Λ a.u T c Ρ 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

31 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 E c Ρ WXC,Λ a.u T c Ρ 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

32 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 E c Ρ WXC,Λ a.u 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

33 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 E c Ρ WXC,Λ a.u T c Ρ Λ A linear curve would reflect a λ 2 dependence of the correlation energy why does the curve bend? Review and preview

34 Two-parameter CCSD correlation curves Second-order Görling Levy theory suggests the following model: E c,λ [ρ] = λ2 g 2 h λg = sλ2 / aλ 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

35 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 E c Ρ WXC,Λ a.u 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

36 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 E c Ρ.5 WXC,Λ a.u W CCSD T Ρ W CCSD Ρ 9.4 T c Ρ CCSD 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

37 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

38 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 HF.9 1. BLYP FCI Dissociation of H 2 and static correlation

39 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 HF BLYP.9 R.7 bohr FCI Dissociation of H 2 and static correlation

40 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 HF BLYP FCI.9 R 1.4 bohr Dissociation of H 2 and static correlation

41 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 HF BLYP FCI R 3. bohr Dissociation of H 2 and static correlation

42 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 R 5. bohr Dissociation of H 2 and static correlation

43 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 R 1. bohr Dissociation of H 2 and static correlation

44 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

45 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

46 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

47 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 R 5. bohr FCI Dissociation of H 2 and static correlation

48 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 R 7. bohr FCI Dissociation of H 2 and static correlation

49 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 R 1. bohr.25 FCI Dissociation of H 2 and static correlation

50 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 Dissociation of H 2 and static correlation

51 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

52 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

53 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

54 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

55 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

56 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

57 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

58 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

59 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

60 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

61 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

62 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

63 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 WΛ,c a.u Generalized adiabatic connections

64 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 WΛ,c a.u Generalized adiabatic connections

65 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 WΛ,c a.u..3.4 WΛ,c a.u Λ Generalized adiabatic connections

66 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 WΛ,c a.u..4 WΛ,c a.u Λ Generalized adiabatic connections

67 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 WΛ,c a.u..4 WΛ,c a.u Λ Generalized adiabatic connections

68 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 WΛ,c a.u..4 WΛ,c a.u Λ Generalized adiabatic connections

69 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 WΛ,c a.u..4 WΛ,c a.u Λ Generalized adiabatic connections

70 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 WΛ,c a.u..4 WΛ,c a.u Λ Generalized adiabatic connections

71 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 WΛ,c a.u..4 WΛ,c a.u Λ Generalized adiabatic connections

72 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 WΛ,c a.u..4 WΛ,c a.u Λ Generalized adiabatic connections

73 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 Λ Λ Generalized adiabatic connections

74 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 WΛ a.u. 1. WΛ a.u Λ. Λ Generalized adiabatic connections

75 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 WΛ a.u. 1. WΛ a.u Λ. Λ Generalized adiabatic connections

76 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 WΛ a.u. 1. WΛ a.u Λ. Λ Generalized adiabatic connections

77 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 WΛ a.u. 1. WΛ a.u Λ. Λ Generalized adiabatic connections

78 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 WΛ a.u. 1. WΛ a.u Λ. Λ Generalized adiabatic connections

79 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 WΛ a.u. 1. WΛ a.u Λ Λ Generalized adiabatic connections

80 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 WΛ,c a.u..2 WΛ,c a.u Λ Static correlation is not easily treated by range separation Generalized adiabatic connections

81 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 WΛ,c a.u..2 WΛ,c a.u Λ Static correlation is not easily treated by range separation Generalized adiabatic connections

82 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 WΛ,c a.u..2 WΛ,c a.u Λ Static correlation is not easily treated by range separation Generalized adiabatic connections

83 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 WΛ,c a.u..2 WΛ,c a.u Λ Static correlation is not easily treated by range separation Generalized adiabatic connections

84 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 WΛ,c a.u..2 WΛ,c a.u Λ Static correlation is not easily treated by range separation Generalized adiabatic connections

85 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 WΛ,c a.u..2 WΛ,c a.u Λ Static correlation is not easily treated by range separation Generalized adiabatic connections

86 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 Λ Generalized adiabatic connections

87 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

Importing ab-initio theory into DFT: Some applications of the Lieb variation principle

Importing ab-initio theory into DFT: Some applications of the Lieb variation principle Importing ab-initio theory into DFT: Some applications of the Lieb variation principle Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department

More information

The calculation of the universal density functional by Lieb maximization

The calculation of the universal density functional by Lieb maximization The calculation of the universal density functional by Lieb maximization Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry,

More information

Ab-initio studies of the adiabatic connection in density-functional theory

Ab-initio studies of the adiabatic connection in density-functional theory Ab-initio studies of the adiabatic connection in density-functional theory Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry,

More information

Density functional theory in magnetic fields

Density functional theory in magnetic fields Density functional theory in magnetic fields U. E. Ekström, T. Helgaker, S. Kvaal, E. Sagvolden, E. Tellgren, A. M. Teale Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry,

More information

Basis sets for electron correlation

Basis sets for electron correlation Basis sets for electron correlation Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 12th Sostrup Summer School Quantum Chemistry

More information

Density-functional theory in quantum chemistry. Trygve Helgaker. From Quarks to the Nuclear Many-Body Problem

Density-functional theory in quantum chemistry. Trygve Helgaker. From Quarks to the Nuclear Many-Body Problem 1 Density-functional theory in quantum chemistry Trygve Helgaker Centre for Theoretical and Computational Chemistry, University of Oslo, Norway From Quarks to the Nuclear Many-Body Problem A conference

More information

Excitation energies from density-functional theory some failures and successes. Trygve Helgaker

Excitation energies from density-functional theory some failures and successes. Trygve Helgaker 1 Excitation energies from density-functional theory some failures and successes Trygve Helgaker Centre for Theoretical and Computational Chemistry, University of Oslo, Norway Ola Lutnæs, Simen Reine,

More information

Density-functional theory

Density-functional theory Density-functional theory Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 13th Sostrup Summer School Quantum Chemistry and Molecular

More information

Highly accurate quantum-chemical calculations

Highly accurate quantum-chemical calculations 1 Highly accurate quantum-chemical calculations T. Helgaker Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo, Norway A. C. Hennum and T. Ruden, University

More information

DFT calculations of NMR indirect spin spin coupling constants

DFT calculations of NMR indirect spin spin coupling constants DFT calculations of NMR indirect spin spin coupling constants Dalton program system Program capabilities Density functional theory Kohn Sham theory LDA, GGA and hybrid theories Indirect NMR spin spin coupling

More information

Short Course on Density Functional Theory and Applications VII. Hybrid, Range-Separated, and One-shot Functionals

Short Course on Density Functional Theory and Applications VII. Hybrid, Range-Separated, and One-shot Functionals Short Course on Density Functional Theory and Applications VII. Hybrid, Range-Separated, and One-shot Functionals Samuel B. Trickey Sept. 2008 Quantum Theory Project Dept. of Physics and Dept. of Chemistry

More information

Orbital dependent correlation potentials in ab initio density functional theory

Orbital dependent correlation potentials in ab initio density functional theory Orbital dependent correlation potentials in ab initio density functional theory noniterative - one step - calculations Ireneusz Grabowski Institute of Physics Nicolaus Copernicus University Toruń, Poland

More information

Molecular Magnetic Properties

Molecular Magnetic Properties Molecular Magnetic Properties Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Raman Centre for Atomic, Molecular and Optical

More information

Computational Methods. Chem 561

Computational Methods. Chem 561 Computational Methods Chem 561 Lecture Outline 1. Ab initio methods a) HF SCF b) Post-HF methods 2. Density Functional Theory 3. Semiempirical methods 4. Molecular Mechanics Computational Chemistry " Computational

More information

Oslo node. Highly accurate calculations benchmarking and extrapolations

Oslo node. Highly accurate calculations benchmarking and extrapolations Oslo node Highly accurate calculations benchmarking and extrapolations Torgeir Ruden, with A. Halkier, P. Jørgensen, J. Olsen, W. Klopper, J. Gauss, P. Taylor Explicitly correlated methods Pål Dahle, collaboration

More information

Extending Kohn-Sham density-functional theory

Extending Kohn-Sham density-functional theory Extending Kohn-Sham density-functional theory Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Email: julien.toulouse@upmc.fr Web page: www.lct.jussieu.fr/pagesperso/toulouse/ January

More information

Théorie de la fonctionnnelle de la densité avec séparation de portée pour les forces de van der Waals

Théorie de la fonctionnnelle de la densité avec séparation de portée pour les forces de van der Waals Théorie de la fonctionnnelle de la densité avec séparation de portée pour les forces de van der Waals Julien Toulouse 1 Iann Gerber 2, Georg Jansen 3, Andreas Savin 1, János Ángyán 4 1 Laboratoire de Chimie

More information

Electronic structure theory: Fundamentals to frontiers. 2. Density functional theory

Electronic structure theory: Fundamentals to frontiers. 2. Density functional theory Electronic structure theory: Fundamentals to frontiers. 2. Density functional theory MARTIN HEAD-GORDON, Department of Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley

More information

T. Helgaker, Department of Chemistry, University of Oslo, Norway. T. Ruden, University of Oslo, Norway. W. Klopper, University of Karlsruhe, Germany

T. Helgaker, Department of Chemistry, University of Oslo, Norway. T. Ruden, University of Oslo, Norway. W. Klopper, University of Karlsruhe, Germany 1 The a priori calculation of molecular properties to chemical accuarcy T. Helgaker, Department of Chemistry, University of Oslo, Norway T. Ruden, University of Oslo, Norway W. Klopper, University of Karlsruhe,

More information

Fundamentals of Density-Functional Theory

Fundamentals of Density-Functional Theory Fundamentals of Density-Functional Theory Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway GdR CORREL Meeting 015 Amphi Sciences Naturelles,

More information

The Accurate Calculation of Molecular Energies and Properties: A Tour of High-Accuracy Quantum-Chemical Methods

The Accurate Calculation of Molecular Energies and Properties: A Tour of High-Accuracy Quantum-Chemical Methods 1 The Accurate Calculation of Molecular Energies and Properties: A Tour of High-Accuracy Quantum-Chemical Methods T. Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry,

More information

Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation

Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation Julien Toulouse 1 I. Gerber 2, G. Jansen 3, A. Savin 1, W. Zhu 1, J. Ángyán 4 1 Laboratoire de Chimie Théorique,

More information

Molecular Magnetism. Magnetic Resonance Parameters. Trygve Helgaker

Molecular Magnetism. Magnetic Resonance Parameters. Trygve Helgaker Molecular Magnetism Magnetic Resonance Parameters Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Laboratoire de Chimie Théorique,

More information

Molecular electronic structure in strong magnetic fields

Molecular electronic structure in strong magnetic fields Trygve Helgaker 1, Mark Hoffmann 1,2 Kai Molecules Lange in strong 1, Alessandro magnetic fields Soncini 1,3, andcctcc Erik Jackson Tellgren 211 1 (CTCC, 1 / 23 Uni Molecular electronic structure in strong

More information

Advanced Quantum Chemistry III: Part 3. Haruyuki Nakano. Kyushu University

Advanced Quantum Chemistry III: Part 3. Haruyuki Nakano. Kyushu University Advanced Quantum Chemistry III: Part 3 Haruyuki Nakano Kyushu University 2013 Winter Term 1. Hartree-Fock theory Density Functional Theory 2. Hohenberg-Kohn theorem 3. Kohn-Sham method 4. Exchange-correlation

More information

Adiabatic connection from accurate wave-function calculations

Adiabatic connection from accurate wave-function calculations JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 12 22 MARCH 2000 Adiabatic connection from accurate wave-function calculations Derek Frydel and William M. Terilla Department of Chemistry, Rutgers University,

More information

The Rigorous Calculation of Molecular Properties to Chemical Accuracy. T. Helgaker, Department of Chemistry, University of Oslo, Norway

The Rigorous Calculation of Molecular Properties to Chemical Accuracy. T. Helgaker, Department of Chemistry, University of Oslo, Norway 1 The Rigorous Calculation of Molecular Properties to Chemical Accuracy T. Helgaker, Department of Chemistry, University of Oslo, Norway A. C. Hennum and T. Ruden, University of Oslo, Norway S. Coriani,

More information

Computational Chemistry I

Computational Chemistry I Computational Chemistry I Text book Cramer: Essentials of Quantum Chemistry, Wiley (2 ed.) Chapter 3. Post Hartree-Fock methods (Cramer: chapter 7) There are many ways to improve the HF method. Most of

More information

The Role of the Hohenberg Kohn Theorem in Density-Functional Theory

The Role of the Hohenberg Kohn Theorem in Density-Functional Theory The Role of the Hohenberg Kohn Theorem in Density-Functional Theory T. Helgaker, U. E. Ekström, S. Kvaal, E. Sagvolden, A. M. Teale,, E. Tellgren Centre for Theoretical and Computational Chemistry (CTCC),

More information

Introduction to density-functional theory. Emmanuel Fromager

Introduction to density-functional theory. Emmanuel Fromager Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS M2 lecture, Strasbourg, France. Institut

More information

1 Density functional theory (DFT)

1 Density functional theory (DFT) 1 Density functional theory (DFT) 1.1 Introduction Density functional theory is an alternative to ab initio methods for solving the nonrelativistic, time-independent Schrödinger equation H Φ = E Φ. The

More information

Ab initio calculations for potential energy surfaces. D. Talbi GRAAL- Montpellier

Ab initio calculations for potential energy surfaces. D. Talbi GRAAL- Montpellier Ab initio calculations for potential energy surfaces D. Talbi GRAAL- Montpellier A theoretical study of a reaction is a two step process I-Electronic calculations : techniques of quantum chemistry potential

More information

Convergence properties of the coupled-cluster method: the accurate calculation of molecular properties for light systems

Convergence properties of the coupled-cluster method: the accurate calculation of molecular properties for light systems 1 Convergence properties of the coupled-cluster method: the accurate calculation of molecular properties for light systems T. Helgaker Centre for Theoretical and Computational Chemistry, Department of

More information

OVERVIEW OF QUANTUM CHEMISTRY METHODS

OVERVIEW OF QUANTUM CHEMISTRY METHODS OVERVIEW OF QUANTUM CHEMISTRY METHODS Outline I Generalities Correlation, basis sets Spin II Wavefunction methods Hartree-Fock Configuration interaction Coupled cluster Perturbative methods III Density

More information

Spring College on Computational Nanoscience May Variational Principles, the Hellmann-Feynman Theorem, Density Functional Theor

Spring College on Computational Nanoscience May Variational Principles, the Hellmann-Feynman Theorem, Density Functional Theor 2145-25 Spring College on Computational Nanoscience 17-28 May 2010 Variational Principles, the Hellmann-Feynman Theorem, Density Functional Theor Stefano BARONI SISSA & CNR-IOM DEMOCRITOS Simulation Center

More information

DFT basée sur le théorème de fluctuation-dissipation avec séparation de portée pour les interactions de van der Waals

DFT basée sur le théorème de fluctuation-dissipation avec séparation de portée pour les interactions de van der Waals DFT basée sur le théorème de fluctuation-dissipation avec séparation de portée pour les interactions de van der Waals Julien Toulouse 1 Iann Gerber 2, Georg Jansen 3, Andreas Savin 1, János Ángyán 4 1

More information

Range-separated density-functional theory with long-range random phase approximation

Range-separated density-functional theory with long-range random phase approximation Range-separated density-functional theory with long-range random phase approximation Julien Toulouse 1 Wuming Zhu 1, Andreas Savin 1, János Ángyán2 1 Laboratoire de Chimie Théorique, UPMC Univ Paris 6

More information

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah 1. Born-Oppenheimer approx.- energy surfaces 2. Mean-field (Hartree-Fock) theory- orbitals 3. Pros and cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usually does HF-how? 6. Basis sets and notations

More information

Molecular Magnetic Properties

Molecular Magnetic Properties Molecular Magnetic Properties Trygve Helgaker Hylleraas Centre, Department of Chemistry, University of Oslo, Norway and Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo,

More information

Molecules in Magnetic Fields

Molecules in Magnetic Fields Molecules in Magnetic Fields Trygve Helgaker Hylleraas Centre, Department of Chemistry, University of Oslo, Norway and Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo, Norway

More information

Molecules in strong magnetic fields

Molecules in strong magnetic fields Molecules in strong magnetic fields Trygve Helgaker, Kai Lange, Alessandro Soncini, and Erik Tellgren Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway

More information

Electron Correlation

Electron Correlation Electron Correlation Levels of QM Theory HΨ=EΨ Born-Oppenheimer approximation Nuclear equation: H n Ψ n =E n Ψ n Electronic equation: H e Ψ e =E e Ψ e Single determinant SCF Semi-empirical methods Correlation

More information

TDDFT in Chemistry and Biochemistry III

TDDFT in Chemistry and Biochemistry III TDDFT in Chemistry and Biochemistry III Dmitrij Rappoport Department of Chemistry and Chemical Biology Harvard University TDDFT Winter School Benasque, January 2010 Dmitrij Rappoport (Harvard U.) TDDFT

More information

XYZ of ground-state DFT

XYZ of ground-state DFT XYZ of ground-state DFT Kieron Burke and Lucas Wagner Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA January 5-9th, 2014 Kieron (UC Irvine) XYZ of ground-state

More information

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah 1. Born-Oppenheimer approx.- energy surfaces 2. Mean-field (Hartree-Fock) theory- orbitals 3. Pros and cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usually does HF-how? 6. Basis sets and notations

More information

Electron Correlation - Methods beyond Hartree-Fock

Electron Correlation - Methods beyond Hartree-Fock Electron Correlation - Methods beyond Hartree-Fock how to approach chemical accuracy Alexander A. Auer Max-Planck-Institute for Chemical Energy Conversion, Mülheim September 4, 2014 MMER Summerschool 2014

More information

Introduction to Computational Chemistry

Introduction to Computational Chemistry Introduction to Computational Chemistry Vesa Hänninen Laboratory of Physical Chemistry Chemicum 4th floor vesa.hanninen@helsinki.fi September 10, 2013 Lecture 3. Electron correlation methods September

More information

Intermediate DFT. Kieron Burke and Lucas Wagner. Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA

Intermediate DFT. Kieron Burke and Lucas Wagner. Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA Intermediate DFT Kieron Burke and Lucas Wagner Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA October 10-19th, 2012 Kieron (UC Irvine) Intermediate DFT Lausanne12

More information

Molecular Magnetic Properties

Molecular Magnetic Properties Molecular Magnetic Properties Trygve Helgaker Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway The 12th Sostrup Summer School Quantum Chemistry and

More information

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah 1. Born-Oppenheimer approx.- energy surfaces 2. Mean-field (Hartree-Fock) theory- orbitals 3. Pros and cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usually does HF-how? 6. Basis sets and notations

More information

Combining density-functional theory and many-body methods

Combining density-functional theory and many-body methods Combining density-functional theory and many-body methods Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Vrije Universiteit Amsterdam, Netherlands November 2017 Outline 2/23 1

More information

QUANTUM CHEMISTRY PROJECT 3: PARTS B AND C

QUANTUM CHEMISTRY PROJECT 3: PARTS B AND C Chemistry 460 Fall 2017 Dr. Jean M. Standard November 6, 2017 QUANTUM CHEMISTRY PROJECT 3: PARTS B AND C PART B: POTENTIAL CURVE, SPECTROSCOPIC CONSTANTS, AND DISSOCIATION ENERGY OF DIATOMIC HYDROGEN (20

More information

Density Functional Theory: from theory to Applications

Density Functional Theory: from theory to Applications Density Functional Theory: from theory to Applications Uni Mainz November 29, 2010 The self interaction error and its correction Perdew-Zunger SIC Average-density approximation Weighted density approximation

More information

Electronic band structure, sx-lda, Hybrid DFT, LDA+U and all that. Keith Refson STFC Rutherford Appleton Laboratory

Electronic band structure, sx-lda, Hybrid DFT, LDA+U and all that. Keith Refson STFC Rutherford Appleton Laboratory Electronic band structure, sx-lda, Hybrid DFT, LDA+U and all that Keith Refson STFC Rutherford Appleton Laboratory LDA/GGA DFT is good but... Naive LDA/GGA calculation severely underestimates band-gaps.

More information

Diamagnetism and Paramagnetism in Atoms and Molecules

Diamagnetism and Paramagnetism in Atoms and Molecules Diamagnetism and Paramagnetism in Atoms and Molecules Trygve Helgaker Alex Borgoo, Maria Dimitrova, Jürgen Gauss, Florian Hampe, Christof Holzer, Wim Klopper, Trond Saue, Peter Schwerdtfeger, Stella Stopkowicz,

More information

Introduction to multiconfigurational quantum chemistry. Emmanuel Fromager

Introduction to multiconfigurational quantum chemistry. Emmanuel Fromager Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS M2 lecture, Strasbourg, France. Notations

More information

Quantum chemistry: wave-function and density-functional methods

Quantum chemistry: wave-function and density-functional methods Quantum chemistry: wave-function and density-functional methods Trygve Helgaker Centre for Theoretical and Computational Chemistry, Department of Chemistry, University of Oslo Electronic Structure e-science

More information

Exercise 1: Structure and dipole moment of a small molecule

Exercise 1: Structure and dipole moment of a small molecule Introduction to computational chemistry Exercise 1: Structure and dipole moment of a small molecule Vesa Hänninen 1 Introduction In this exercise the equilibrium structure and the dipole moment of a small

More information

Session 1. Introduction to Computational Chemistry. Computational (chemistry education) and/or (Computational chemistry) education

Session 1. Introduction to Computational Chemistry. Computational (chemistry education) and/or (Computational chemistry) education Session 1 Introduction to Computational Chemistry 1 Introduction to Computational Chemistry Computational (chemistry education) and/or (Computational chemistry) education First one: Use computational tools

More information

Exchange Correlation Functional Investigation of RT-TDDFT on a Sodium Chloride. Dimer. Philip Straughn

Exchange Correlation Functional Investigation of RT-TDDFT on a Sodium Chloride. Dimer. Philip Straughn Exchange Correlation Functional Investigation of RT-TDDFT on a Sodium Chloride Dimer Philip Straughn Abstract Charge transfer between Na and Cl ions is an important problem in physical chemistry. However,

More information

Time-independent molecular properties

Time-independent molecular properties Time-independent molecular properties Trygve Helgaker Hylleraas Centre, Department of Chemistry, University of Oslo, Norway and Centre for Advanced Study at the Norwegian Academy of Science and Letters,

More information

Molecules in strong magnetic fields

Molecules in strong magnetic fields Molecules in strong magnetic fields Trygve Helgaker, Kai Lange, Alessandro Soncini, and Erik Tellgren Centre for Theoretical and Computational Chemistry Department of Chemistry, University of Oslo, Norway

More information

Introduction to Computational Chemistry: Theory

Introduction to Computational Chemistry: Theory Introduction to Computational Chemistry: Theory Dr Andrew Gilbert Rm 118, Craig Building, RSC andrew.gilbert@anu.edu.au 3023 Course Lectures Introduction Hartree Fock Theory Basis Sets Lecture 1 1 Introduction

More information

Wave function methods for the electronic Schrödinger equation

Wave function methods for the electronic Schrödinger equation Wave function methods for the electronic Schrödinger equation Zürich 2008 DFG Reseach Center Matheon: Mathematics in Key Technologies A7: Numerical Discretization Methods in Quantum Chemistry DFG Priority

More information

Lec20 Fri 3mar17

Lec20 Fri 3mar17 564-17 Lec20 Fri 3mar17 [PDF]GAUSSIAN 09W TUTORIAL www.molcalx.com.cn/wp-content/uploads/2015/01/gaussian09w_tutorial.pdf by A Tomberg - Cited by 8 - Related articles GAUSSIAN 09W TUTORIAL. AN INTRODUCTION

More information

Molecular Magnetic Properties. The 11th Sostrup Summer School. Quantum Chemistry and Molecular Properties July 4 16, 2010

Molecular Magnetic Properties. The 11th Sostrup Summer School. Quantum Chemistry and Molecular Properties July 4 16, 2010 1 Molecular Magnetic Properties The 11th Sostrup Summer School Quantum Chemistry and Molecular Properties July 4 16, 2010 Trygve Helgaker Centre for Theoretical and Computational Chemistry, Department

More information

Teoría del Funcional de la Densidad (Density Functional Theory)

Teoría del Funcional de la Densidad (Density Functional Theory) Teoría del Funcional de la Densidad (Density Functional Theory) Motivation: limitations of the standard approach based on the wave function. The electronic density n(r) as the key variable: Functionals

More information

MO Calculation for a Diatomic Molecule. /4 0 ) i=1 j>i (1/r ij )

MO Calculation for a Diatomic Molecule. /4 0 ) i=1 j>i (1/r ij ) MO Calculation for a Diatomic Molecule Introduction The properties of any molecular system can in principle be found by looking at the solutions to the corresponding time independent Schrodinger equation

More information

Exchange-Correlation Functional

Exchange-Correlation Functional Exchange-Correlation Functional Aiichiro Nakano Collaboratory for Advanced Computing & Simulations Depts. of Computer Science, Physics & Astronomy, Chemical Engineering & Materials Science, and Biological

More information

Density Functional Theory. Martin Lüders Daresbury Laboratory

Density Functional Theory. Martin Lüders Daresbury Laboratory Density Functional Theory Martin Lüders Daresbury Laboratory Ab initio Calculations Hamiltonian: (without external fields, non-relativistic) impossible to solve exactly!! Electrons Nuclei Electron-Nuclei

More information

Independent electrons in an effective potential

Independent electrons in an effective potential ABC of DFT Adiabatic approximation Independent electrons in an effective potential Hartree Fock Density Functional Theory MBPT - GW Density Functional Theory in a nutshell Every observable quantity of

More information

Generalized generalized gradient approximation: An improved density-functional theory for accurate orbital eigenvalues

Generalized generalized gradient approximation: An improved density-functional theory for accurate orbital eigenvalues PHYSICAL REVIEW B VOLUME 55, NUMBER 24 15 JUNE 1997-II Generalized generalized gradient approximation: An improved density-functional theory for accurate orbital eigenvalues Xinlei Hua, Xiaojie Chen, and

More information

Density Functional Theory - II part

Density Functional Theory - II part Density Functional Theory - II part antonino.polimeno@unipd.it Overview From theory to practice Implementation Functionals Local functionals Gradient Others From theory to practice From now on, if not

More information

Dispersion energies from the random-phase approximation with range separation

Dispersion energies from the random-phase approximation with range separation 1/34 Dispersion energies from the random-phase approximation with range separation Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Web page: www.lct.jussieu.fr/pagesperso/toulouse/

More information

3/23/2010 More basics of DFT Kieron Burke and friends UC Irvine Physics and Chemistry References for ground-state DFT ABC of DFT, by KB and Rudy Magyar, http://dft.uci.edu A Primer in Density Functional

More information

Electric properties of molecules

Electric properties of molecules Electric properties of molecules For a molecule in a uniform electric fielde the Hamiltonian has the form: Ĥ(E) = Ĥ + E ˆµ x where we assume that the field is directed along the x axis and ˆµ x is the

More information

A new generation of density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem

A new generation of density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem A new generation of density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem Andreas Görling, Patrick Bleiziffer, Daniel Schmidtel, and Andreas Heßelmann Lehrstuhl für

More information

Introduction to Computational Quantum Chemistry: Theory

Introduction to Computational Quantum Chemistry: Theory Introduction to Computational Quantum Chemistry: Theory Dr Andrew Gilbert Rm 118, Craig Building, RSC 3108 Course Lectures 2007 Introduction Hartree Fock Theory Configuration Interaction Lectures 1 Introduction

More information

QUANTUM CHEMISTRY FOR TRANSITION METALS

QUANTUM CHEMISTRY FOR TRANSITION METALS QUANTUM CHEMISTRY FOR TRANSITION METALS Outline I Introduction II Correlation Static correlation effects MC methods DFT III Relativity Generalities From 4 to 1 components Effective core potential Outline

More information

Density functional theory in the solid state

Density functional theory in the solid state Density functional theory in the solid state Ari P Seitsonen IMPMC, CNRS & Universités 6 et 7 Paris, IPGP Department of Applied Physics, Helsinki University of Technology Physikalisch-Chemisches Institut

More information

QUANTUM CHEMISTRY PROJECT 3: ATOMIC AND MOLECULAR STRUCTURE

QUANTUM CHEMISTRY PROJECT 3: ATOMIC AND MOLECULAR STRUCTURE Chemistry 460 Fall 2017 Dr. Jean M. Standard November 1, 2017 QUANTUM CHEMISTRY PROJECT 3: ATOMIC AND MOLECULAR STRUCTURE OUTLINE In this project, you will carry out quantum mechanical calculations of

More information

Supporting Information: Predicting the Ionic Product of Water

Supporting Information: Predicting the Ionic Product of Water Supporting Information: Predicting the Ionic Product of Water Eva Perlt 1,+, Michael von Domaros 1,+, Barbara Kirchner 1, Ralf Ludwig 2, and Frank Weinhold 3,* 1 Mulliken Center for Theoretical Chemistry,

More information

Introduction to computational chemistry Exercise I: Structure and electronic energy of a small molecule. Vesa Hänninen

Introduction to computational chemistry Exercise I: Structure and electronic energy of a small molecule. Vesa Hänninen Introduction to computational chemistry Exercise I: Structure and electronic energy of a small molecule Vesa Hänninen 1 Introduction In this exercise the equilibrium structure and the electronic energy

More information

Optimization of quantum Monte Carlo wave functions by energy minimization

Optimization of quantum Monte Carlo wave functions by energy minimization Optimization of quantum Monte Carlo wave functions by energy minimization Julien Toulouse, Roland Assaraf, Cyrus J. Umrigar Laboratoire de Chimie Théorique, Université Pierre et Marie Curie and CNRS, Paris,

More information

Density Functional Theory

Density Functional Theory Chemistry 380.37 Fall 2015 Dr. Jean M. Standard October 28, 2015 Density Functional Theory What is a Functional? A functional is a general mathematical quantity that represents a rule to convert a function

More information

Differentiable but exact formulation of density-functional theory

Differentiable but exact formulation of density-functional theory Differentiable but exact formulation of density-functional theory Simen Kvaal, 1, a) Ulf Ekström, 1 Andrew M. Teale, 2, 1 and Trygve Helgaker 1 1) Centre for Theoretical and Computational Chemistry, Department

More information

Divergence in Møller Plesset theory: A simple explanation based on a two-state model

Divergence in Møller Plesset theory: A simple explanation based on a two-state model JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 22 8 JUNE 2000 Divergence in Møller Plesset theory: A simple explanation based on a two-state model Jeppe Olsen and Poul Jørgensen a) Department of Chemistry,

More information

Introduction to Computational Chemistry Computational (chemistry education) and/or. (Computational chemistry) education

Introduction to Computational Chemistry Computational (chemistry education) and/or. (Computational chemistry) education Introduction to Computational Chemistry Computational (chemistry education) and/or (Computational chemistry) education First one: Use computational tools to help increase student understanding of material

More information

Density Func,onal Theory (Chapter 6, Jensen)

Density Func,onal Theory (Chapter 6, Jensen) Chem 580: DFT Density Func,onal Theory (Chapter 6, Jensen) Hohenberg- Kohn Theorem (Phys. Rev., 136,B864 (1964)): For molecules with a non degenerate ground state, the ground state molecular energy and

More information

Quantum Chemistry Methods

Quantum Chemistry Methods 1 Quantum Chemistry Methods T. Helgaker, Department of Chemistry, University of Oslo, Norway The electronic Schrödinger equation Hartree Fock theory self-consistent field theory basis functions and basis

More information

QMC dissociation energy of the water dimer: Time step errors and backflow calculations

QMC dissociation energy of the water dimer: Time step errors and backflow calculations QMC dissociation energy of the water dimer: Time step errors and backflow calculations Idoia G. de Gurtubay and Richard J. Needs TCM group. Cavendish Laboratory University of Cambridge Idoia G. de Gurtubay.

More information

Periodic Trends in Properties of Homonuclear

Periodic Trends in Properties of Homonuclear Chapter 8 Periodic Trends in Properties of Homonuclear Diatomic Molecules Up to now, we have discussed various physical properties of nanostructures, namely, two-dimensional - graphene-like structures:

More information

Walter Kohn was awarded with the Nobel Prize in Chemistry in 1998 for his development of the density functional theory.

Walter Kohn was awarded with the Nobel Prize in Chemistry in 1998 for his development of the density functional theory. Walter Kohn was awarded with the Nobel Prize in Chemistry in 1998 for his development of the density functional theory. Walter Kohn receiving his Nobel Prize from His Majesty the King at the Stockholm

More information

Module 6 1. Density functional theory

Module 6 1. Density functional theory Module 6 1. Density functional theory Updated May 12, 2016 B A DDFT C K A bird s-eye view of density-functional theory Authors: Klaus Capelle G http://arxiv.org/abs/cond-mat/0211443 R https://trac.cc.jyu.fi/projects/toolbox/wiki/dft

More information

计算物理作业二. Excercise 1: Illustration of the convergence of the dissociation energy for H 2 toward HF limit.

计算物理作业二. Excercise 1: Illustration of the convergence of the dissociation energy for H 2 toward HF limit. 计算物理作业二 Excercise 1: Illustration of the convergence of the dissociation energy for H 2 toward HF limit. In this exercise, basis indicates one of the following basis sets: STO-3G, cc-pvdz, cc-pvtz, cc-pvqz

More information

Basics of DFT. Kieron Burke and Lucas Wagner. Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA

Basics of DFT. Kieron Burke and Lucas Wagner. Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA Basics of DFT Kieron Burke and Lucas Wagner Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA October 10-19th, 2012 Kieron (UC Irvine) Basics of DFT Lausanne12 1

More information

Molecular Magnetic Properties

Molecular Magnetic Properties Molecular Magnetic Properties Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway European Summer School in Quantum Chemistry

More information

Chemical bonding in strong magnetic fields

Chemical bonding in strong magnetic fields Trygve Helgaker 1, Mark Hoffmann 1,2 Chemical Kai Lange bonding in 1, strong Alessandro magnetic fields Soncini 1,3, CMS212, and Erik JuneTellgren 24 27 212 1 (CTCC, 1 / 32 Uni Chemical bonding in strong

More information

JULIEN TOULOUSE, PAOLA GORI-GIORGI, ANDREAS SAVIN

JULIEN TOULOUSE, PAOLA GORI-GIORGI, ANDREAS SAVIN Scaling Relations, Virial Theorem, and Energy Densities for Long-Range and Short-Range Density Functionals JULIEN TOULOUSE, PAOLA GORI-GIORGI, ANDREAS SAVIN Laboratoire de Chimie Théorique, CNRS et Université

More information