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PSU CHEM 408 - Post HF Calculations

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2/17/2005 CHEM 408 - Sp05L6 - 1• HF calculations typically account for ~99% of Eel; unfortunately that’s not good enough for many chemical problems• post-HF methods attempt to recover the missing “correlation energy”, Ecorr= Eel- EHF, by improving on ΨHF• HF-SCF results provide the best Ψ and lowest Eelpossible consistent with the orbital approximation• Veeis treated approximately - each e feels only the average charge distribution created by the remaining es, not the direct rij-1interactions actually present in Ĥ“Post HF” Calculations“Post HF” Calculations......)3,2,1(321χχχ=Ψ• we’ll consider 3 post-HF approaches:CI - configuration interaction (CIS, CISD, QCISD, …)MBPT - many-body perturbation theory (MP2, MP3…)(CC - coupled cluster theory)2/17/2005 CHEM 408 - Sp05L6 - 2• the problem with HF Ψs is illustrated by the dissociation of H2:Figure from F. Jensen, Introduction to Computational Chemistry (Wiley, 1999).exact (full CI)RHFUHF (a partial fix)2/17/2005 CHEM 408 - Sp05L6 - 3• consider minimal basis set HF calculation of H2:basis set: {1sA, 1sB}MOs)]1(1)1(1[)1()]1(1)1(1[)1(BAuBAgssNssN−′=+=σσ•at any RABthe ground state is described by (ignore Ns.):• expand spatial part:)}2()1()2()1(){2()1( )2()1()2()1()2,1(2αββασσσσσσσ−=−==Ψgggggggσgσu)2(1)1(1)2(1)1(1)2(1)1(1)2(1)1(1)2()1(ABBABBAAggssssssss+++=σσ“ionic” H++ H-“covalent” H· + H· ABABABAB• although all four terms contribute for RAB~Re, the ionic contributions should be negligible as RAB→∞2/17/2005 CHEM 408 - Sp05L6 - 4configuration interaction (CI) calculations mix HF Ψ with excited Ψs in a variational calculation to allow for e-e correlation:σgσuΨHF2gσugσσΨ = c0 Ψ0+ c1 “Ψ1” + c2 “Ψ2” + … with variable c0, c1, …2uσsinglyexciteddeterms.doublyexciteddeterms.+ …multiplyexciteddeterms.• full CI allows for mixing of all possible configurations = distributions of electrons among orbitals• in the limit of ∞ basis set size, a full CI calc is exact2/17/2005 CHEM 408 - Sp05L6 - 5• unfortunately, full CI is impossible for all but the smallest systems• for K basis funcs. (2K spin-orbitals) and n electrons,!)2()!2(!)!2(nKnKnKnnK⎯⎯→⎯−=>># Slater determs.H-F Benzenen =10 n =42Basis K # det Basis K # detSTO-3G 6 66 STO-3G 36 1.7E+213-21G 9 2.3E+05 3-21G 66 7.2E+356-31G(d,p) 18 1.7E+09 6-31G(d,p) 120 2.2E+48cc-pVTZ 44 3.4E+13 cc-pVTZ 264 4.7E+63• although ~109determinants can be optimized, it is usually better to limit the CI calculation in some wayso, for example2/17/2005 CHEM 408 - Sp05L6 - 6E……………………HOMOLUMO………………CISCISD……only gs.configurationall singleexcitationsall doubleexcitationsall excitationsin active spaceCAS(SCF)HFCISDTQ… all single, double, triple, quadruple… excitations QCISD quadratic single + double CI method2/17/2005 CHEM 408 - Sp05L6 - 7E……RHFUHF RHFUHF…………• RHF = (spin) restricted HF - closed shell systems near equilibrium• UHF = (spin) unrestricted HF - open shell systems, diradicalsexact α, βMOsidenticalα, βMOstreated separately2/17/2005 CHEM 408 - Sp05L6 - 8Many-Body (or Møller-Plesset) Perturbation Theory• write Ĥ as “easy” and “hard” parts:VHHˆˆˆ0λ+=• Ĥ0corresponds to the “HF” Ĥ (for which ΨHFis exact):∑∑>Ψ⎟⎟⎠⎞⎜⎜⎝⎛−−−=><−=iijijijjieeeeKJrrVVVHF MOoccupied MOoccupied]2[||12ˆˆrr• the perturbation solution can be carried out to different orders in λ:......)3(3)2(2)1(1)0(0)3(3)2(2)1(1)0(0+Ψ+Ψ+Ψ+Ψ=Ψ++++=λλλλλλλλEEEEEwith the higher orders adding smaller and smaller corrections• as in CI methods, the Ψ corrections Ψ(1), Ψ(2), … are sums over excited determinants Coulomb & exchange integrals from HF theoryexact Vee2/17/2005 CHEM 408 - Sp05L6 - 9• the 1st correction to the HF solution is at 2nd order, where theenergy correction is:∑∑∑∑∫∫>>−−+−−=iijaabjibaabbajirdrdrrEεεεεψψψψψψ2121)2()]2()1()2()1([||1)2()1(rrrroccupiedMOs i, junoccupiedMOs i, jMO energies• this 2nd order solution is termed “MP2” and higher order corrections are “MPn”, with n=3, 4, …• an MP2 correction is often comparable in expense to the original HF calculation; it captures 80-90% of the correlation energy and is therefore a good choice for many applications•MPn not variational; often oscillatorypropertyHFMP2MP3MP42/17/2005 CHEM 408 - Sp05L6 - 10(aside) Coupled Cluster (CC) Theory• defined by the formal expressions: HFT Ψ=Ψ )ˆexp(electrons) # ( ˆ...ˆˆˆˆ321=+++= nTTTTTnandwhere Tioperators create all possible ithorder excitations from ΨHF(i.e. T1generates single excitations, T2double excitations, etc.)• like full CI, use of complete T is exact but usually impossible• truncated versions of method (e.g. CCSD where one assumes T ≅T1+ T2) are analogous to truncated CI methods but are more accurate (and more expensive)• CCSDT is one of the most accurate methods available, but it can be applied to small molecules; CCSD(T) cheaper & almost as good• CCSD is virtually identical to QCISD (and QCISD(T) ≅ CCSD(T))2/17/2005 CHEM 408 - Sp05L6 - 11Some Method Comparisonsvariational - does lower energy mean more accurate?size consistent - does calculation scale properly with system size?Method variational? size consistent?HF y y full CI y y CIS, CISD, … y n MPn n y CC n y • two key characteristics:2/17/2005 CHEM 408 - Sp05L6 - 12Accuracy: Correlation and Basis SetFigure adapted from Foresman& Frisch, Exploring Chemistry(Gaussian, 1993).Basis Set CompletenessCorrelation Accuracyexactsoln.HFlimit∞2/17/2005 CHEM 408 - Sp05L6 - 13Basis Set RecommendationsTable from:Foresman & Frisch, Exploring Chemistry (Gaussian, 1993).2/17/2005 CHEM 408 - Sp05L6 - 14• accuracy with medium size basis sets is often in the order*:HF << MP2 < CISD < CCSD < MP4 < CCSD(T)* F. Jensen, Introduction to Computational Chemistry (Wiley, 1999).• applicability for single-point calculations (DZP type basis set)*:Nb4Nb5Nb6Nb7accuracyscaling = expenseNb= # basis funcs.HF Nb< 5000 < 200 CH2MP2 Nb< 800 < 30 CH2CCSD(T) Nb< 300 < 10 CH2• for geometry optimizations these sizes should be halved• Nbscaling makes approach to exact limit unfeasible in most cases2/17/2005 CHEM 408 - Sp05L6 - 15Some Relative TimingsCH4n-C5H12Tables from:Foresman & Frisch, Exploring Chemistry (Gaussian, 1993).!2/17/2005 CHEM 408 - Sp05L6 - 16Multi-Level (or Compound)

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