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 and Molecular Properties July 1 13 2012 Trygve Helgaker (CTCC, University of Oslo) Basis setes for electron correlation 12th Sostrup Summer School (2012) 1 / 24

Introduction Requirements for correlated and uncorrelated wave-function models are different uncorrelated models require an accurate representation of the one-electron density correlated models require also an accurate representation of the two-electron density We have discussed basis functions and basis sets for uncorrelated methods we are now going to consider basis set for electron correlation Overview 1 the Coulomb hole and Coulomb cusp 2 basis-set convergence of the correlation energy conventional CI explicitly correlated R12-CI the Hylleraas function 3 partial-wave and principal expansions 4 atomic natural orbitals 5 correlation-consistent basis sets 6 basis-set extrapolation Trygve Helgaker (CTCC, University of Oslo) Overview 12th Sostrup Summer School (2012) 2 / 24

The local kinetic energy Consider the local energy of the helium atom E loc = (HΨ)/Ψ constant for the exact wave function The electronic Hamiltonian has singularities at points of coalescence H = 1 2 2 1 1 2 2 2 2 r 1 2 r 2 + 1 r 12 infinite potential terms canceled by infinite kinetic terms at coalescence 300 Local kinetic energy in the helium atom positive around the nucleus negative around the second electron 200 Negative kinetic energy counterintuitive classical forbidden region 0 internal tunneling w. f. decays towards the singularity 100 the Coulomb hole 0.5 100 0.0 0.5 0.5 1.0 0.0 Trygve Helgaker (CTCC, University of Oslo) The Coulomb cusp and Coulomb hole 12th Sostrup Summer School (2012) 3 / 24

The Coulomb hole Each electron is surrounded by a classically forbidden region: the Coulomb hole without a good description of this region, our results will be inaccurate The helium wave function with one electron fixed at a separation of 0.5a 0 from the nucleus total wave function with the corresponding Hartree Fock wave function subtracted 0.5 0.0-0.5 0.00-0.05-1.0-0.10-0.5 0.0 0.5 1.0 Trygve Helgaker (CTCC, University of Oslo) The Coulomb cusp and Coulomb hole 12th Sostrup Summer School (2012) 4 / 24

Cusp conditions Consider the helium Hamiltonian expressed in terms of r 1, r 2, and r 12 : ( ) H = 1 2 2 2 r 2 + 2 + 2Z ( 2 i=1 i r i r i r i r12 2 + 2 1 ) + r 12 r 12 r 12 The nuclear cusp condition at r i = 0: ( 2 + 2Z ) Ψ r i r i r i ri =0 = 0 Ψ r i = ZΨ (r i = 0) ri =0 easy to satisfy by the use of STOs The Coulomb cusp condition at r 12 = 0: ( 2 1 ) Ψ r 12 r 12 r = 0 Ψ 12 r12 =0 r = 1 12 r12 =0 2 Ψ (r 12 = 0) we shall see that this cannot be satisfied by orbital-based wave functions 0.4 0.1 1.0 0.5 0.5 1.0 Trygve Helgaker (CTCC, University of Oslo) The Coulomb cusp and Coulomb hole 12th Sostrup Summer School (2012) 5 / 24

Convergence of the helium ground-state energy The short-range interactions are difficult to describe we must model the hole accurately for chemical accuracy in our calculations We shall compare the convergence of the following expansions for the helium ground-state 1 conventional CI single-zeta STOs numerical orbitals 2 CI with a correlating function CI-R12 3 the Hylleraas function 0.4 0.1 1.0 0.5 0.5 1.0 Trygve Helgaker (CTCC, University of Oslo) Convergence of the helium ground-state energy 12th Sostrup Summer School (2012) 6 / 24

Configuration-interaction wave function for helium Our one-electron basis functions are STOs: χ nlm (r, θ, ϕ) = r n 1 exp ( ζr) Y lm (θ, ϕ) 2l + 1 (l m)! Y lm (θ, ϕ) = 4π (l + m)! Pm l (cos θ) eimϕ the associated Legendre polynomials Pl m (x) are orthogonal on [ 1, 1] The helium ground-state FCI wave function constructed from such STOs becomes: Ψ FCI (r 1, r 2 ) = exp [ ζ (r 1 + r 2 )] Pl 0 (cos θ 12) ( ) r n 1 1 1 r n 2 1 2 + r n 1 1 2 r n 2 1 1 l n 1 n 2 we have here used the addition theorem Pl 0 (cos θ 12) = 4π l ( 1) m Y l,m (θ 1, ϕ 1 )Y l, m (θ 2, ϕ 2 ) 2l + 1 l Note: the CI expansion uses only three coordinates: r 1, r 2, cos θ 12 the interelectronic distance r 12 does not enter directly Trygve Helgaker (CTCC, University of Oslo) Convergence of the helium ground-state energy 12th Sostrup Summer School (2012) 7 / 24

The principal expansion Include in the FCI wave function all STOs up to a given principal quantum number: N = 1 : Ψ 1 = 1s 2 N = 2 : Ψ 2 = c 1 1s 2 + c 2 1s2s + c 3 2s 2 + c 4 2p 2 The principal expansion converges very slowly it is difficult to obtain an error smaller than 0.1 me h 50 100 150 200 0 10 2 10 4 singleζ CI numerical CI The use of fully numerical orbitals reduces the error by a few factors it does not improve on the intrinsically slow FCI convergence Trygve Helgaker (CTCC, University of Oslo) Convergence of the helium ground-state energy 12th Sostrup Summer School (2012) 8 / 24

Correlating functions By introducing cos θ 12 = r 12 2 r 1 2 r 2 2 2r 1 r 2 obtained from r 12 r 12, we may write the FCI wave function in the form Ψ FCI (r 1, r 2, r 12 ) = exp [ ζ (r 1 + r 2 )] ( ) c ijk r i ijk 1 r j 2 + r 2 i r j 1 r12 2k Since only even powers of r 12 are included, the cusp condition can never be satisfied Ψ CI r = 0 12 r12 =0 However, if we include a term linear in r 12 Ψ CI r 12 = ( 1 + 1 2 r 12 ) Ψ CI then the cusp condition is satisfied exactly Ψ CI r 12 = 1 r 12 2 ΨCI (r 12 = 0) = 1 2 ΨCI r 12 (r 12 = 0) r12 =0 We may always satisfy the cusp condition by multiplication with a correlating function: γ = 1 + 1 2 r ij i>j Trygve Helgaker (CTCC, University of Oslo) Convergence of the helium ground-state energy 12th Sostrup Summer School (2012) 9 / 24

Explicitly correlated methods Methods that employ correlating functions or otherwise make explicit use of the interelectronic distances r ij are known as explicitly correlated methods the R12 method includes r ij linearly the F12 method includes a more general (exponential) dependence on r ij The R12 principal expansion Ψ R12 N = ΨCI N + c 12r 12 Ψ CI 1 converges easily to within 0.1 me h (chemical accuracy) 50 100 150 200 10 2 10 4 singleζ CI numerical CI 10 6 singleζ CIR12 Still, it appears difficult to converge to within 1 µe h (spectroscopic accuracy) Trygve Helgaker (CTCC, University of Oslo) Convergence of the helium ground-state energy 12th Sostrup Summer School (2012) 10 / 24

The Hylleraas function Finally, we include in the wave function all powers of r 12 Ψ H (r 1, r 2, r 12 ) = exp [ ζ (r 1 + r 2 )] c ijk (r 1 i r j 2 + r 2 i r j 1 r12 k ijk This wave function is usually expressed in terms of the Hylleraas coordinates s = r 1 + r 2, t = r 1 r 2, u = r 12 Only even powers in t are needed for the singlet ground state: ) Ψ H (r 1, r 2, r 12 ) = exp ( ζs) ijk c ijk s i t 2j u k The Hylleraas function converges easily to within 0.1 µe h 50 100 150 200 0 10 2 10 4 10 6 10 8 singleζ CI numerical CI singleζ CIR12 Hylleraas The Hylleraas method cannot easily be generalized to many-electron systems Trygve Helgaker (CTCC, University of Oslo) Convergence of the helium ground-state energy 12th Sostrup Summer School (2012) 11 / 24

Convergence rates We have seen the reason for the slow convergence of FCI wave functions DZ TZ QZ 5Z -90 90-90 90-90 90-90 90 Let us now examine the rate of convergence for the helium atom using the 1 partial-wave expansion 2 principal expansion 4f 3d 4d 2p 3p 4p 1s 2s 3s 4s 4f 5f 6f 3d 4d 5d 6d 2p 3p 4p 5p 6p 1s 2s 3s 4s 5s 6s Trygve Helgaker (CTCC, University of Oslo) The partial-wave and principal expansions 12th Sostrup Summer School (2012) 12 / 24

The partial-wave expansion of helium Consider the expansion of the helium FCI wave function in partial waves: L Ψ CI L = ψ l l=0 this expansion has been studied in great detail theoretically Each partial wave is an infinite expansion in determinants it contains all possible combinations of orbitals of angular momentum l, for example 1s 2, 1s2s, 2s 2, 1s3s, 2s3s, 3s 2,... The contribution from each partial wave converges asymptotically as E L = E L E L 1 = 0.074226 ( L + 1 ) 4 ( ) 2 0.030989 L + 1 5 2 + Convergence is slow but systematic Trygve Helgaker (CTCC, University of Oslo) The partial-wave and principal expansions 12th Sostrup Summer School (2012) 13 / 24

The principal expansion of helium The partial-wave expansion is difficult to realize in practice The alternative principal expansion contains a finite number of terms at each level Ψ 1 : 1s 2 Ψ 2 : 1s 2, 1s2s, 2s 2, 2p 2 The principal expansion is higher in energy at each truncation level (E h ): L E L N E N 0 2.879 1 2.862 1 2.901 2 2.898 2 2.903 3 2.902 3 2.904 4 2.903 However, the asymptotic convergence rate of the energy corrections is the same E N = E N E N 1 = c 4 ( N 1 2 ) 4 + Trygve Helgaker (CTCC, University of Oslo) The partial-wave and principal expansions 12th Sostrup Summer School (2012) 14 / 24

Energy contributions and errors The contribution to the correlation energy from each AO in large helium CI calculations is E nlm = an 6 E nlm = π4 90 a = 1.08a nlm The contribution from each partial wave is therefore: E l = a (2l + 1) n 6 a (2l + 1) n 6 dn n=l+1 l+1/2 = 1 5 a(2l + 1) ( l + 1 ) 5 2 = 2 5 a ( l + 1 ) 4 2 The asymptotic truncation error of the partial-wave expansion with l L is therefore E L = E L E = 2 5 a ( ) l + 1 4 2 + 2 5 a ( ) l + 1 4 2 dl = 2 a (L + 1) 3 15 l=l+1 L+1/2 The contribution from each shell in the principal expansion is: E n = an 2 n 6 = an 4 The asymptotic truncation error of the principal expansion with n N is therefore E N = E N E = a n 4 a n 4 dn = 1 3 a(n + 1 2 ) 3 n=n+1 N+1/2 The two series converge slowly but smoothly and may therefore be extrapolated Trygve Helgaker (CTCC, University of Oslo) The partial-wave and principal expansions 12th Sostrup Summer School (2012) 15 / 24

Some observations The number of AOs at truncation level N in the principal expansion is given by N N ao = n 2 = 1 N(N + 1)(N + 2) N3 6 i=1 It follows that the error is inversely proportional to the number of AOs: E N N 3 N 1 ao The dependence of the error in the correlation energy on the CPU time is thus: E N T 1/4 Each new digit in the energy therefore costs 10000 times more CPU time! The convergence is exceedingly slow! 1 minute 1 week 200 years A brute-force basis-set extension until convergence may not always be possible. Fortunately, the convergence is very smooth, allowing for extrapolation. Trygve Helgaker (CTCC, University of Oslo) The partial-wave and principal expansions 12th Sostrup Summer School (2012) 16 / 24

Basis sets for correlated calculations We must provide correlating orbitals for the virtual space The requirements are more severe than for uncorrelated calculations Expect slow but systematic convergence for the description of short-range interactions Overview 1 valence and core-valence correlation 2 atomic natural orbitals (ANOs) 3 correlation-consistent basis sets 4 basis-set extrapolation Trygve Helgaker (CTCC, University of Oslo) Basis sets for correlated calculations 12th Sostrup Summer School (2012) 17 / 24

Valence and core correlation The core electrons are least affected by chemical processes For many purposes, it is sufficient to correlate the valence electrons Example: the dissociation of BH to the left, total electronic energies to the right, core and valence correlation energies Ecore corr = E all corr Eval corr 25.0 HF core FCI fc valence 25.2 FCI all 1 2 3 4 5 1 2 3 4 5 The valence correlation energy can be recovered with smaller basis sets Trygve Helgaker (CTCC, University of Oslo) Basis sets for correlated calculations 12th Sostrup Summer School (2012) 18 / 24

Atomic natural orbitals (ANOs) ANOs are obtained by diagonalizing the one-electron CISD atomic density matrix We obtain a large primitive basis that is generally contracted The ANOs constitute a hierarchical basis of the same structure as the principal expansion 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 5g The occupation numbers provide a natural criterion for selecting basis functions: s p d f g η 1l 2.000000 η 2l 1.924675 0.674781 η 3l 0.008356 0.004136 0.008834 η 4l 0.000347 0.000331 0.000124 0.000186 η 5l 0.000021 0.000034 0.000016 0.000011 0.000018 Trygve Helgaker (CTCC, University of Oslo) Basis sets for correlated calculations 12th Sostrup Summer School (2012) 19 / 24

Correlation-consistent basis sets The correlation-consistent basis sets constitute a realization of the principal expansion: 1 begin with a generally contracted set of atomic Hartree Fock orbitals 2 add primitive energy-optimized correlating orbitals, one shell at a time The resulting correlation-consistent basis sets forms a hierarchical system: cc-pvx Z, X is the cardinal number SZ cc-pvdz cc-pvtz cc-pvqz number of AOs +3s3p3d +4s4p4d4f +5s5p5d5f5g X 2 2s1p 3s2p1d 4s3p2d1f 5s4p3d2f1g X 3 The number of basis functions is given by N X = 1 (X + 1)(X + 3/2)(X + 2) 3 The proportion of the correlation energy recovered increases slowly: Extensions: X 2 3 4 5 6 % 67 88 95 97 98 aug-cc-pvx Z, cc-pcvx Z, aug-cc-pcvx Z Trygve Helgaker (CTCC, University of Oslo) Basis sets for correlated calculations 12th Sostrup Summer School (2012) 20 / 24

cc-pvx Z basis sets cc-pvdz: 3s2p1d 2 s 2 p 2 d 1 1 1 cc-pvtz: 4s3p2d1f s p d 2 1 f cc-pvqz: 5s4p3d2f1g s p d 2 f 2 g 1 1 Trygve Helgaker (CTCC, University of Oslo) Basis sets for correlated calculations 12th Sostrup Summer School (2012) 21 / 24

Correlation-consistent basis sets Percentage of correlation energy recovered with standard and numerical orbitals: X 2 3 4 5 cc-pvdz 77.1 93.0 97.3 98.7 numerical 85.6 95.6 98.0 98.9 The Coulomb hole calculated with standard cc-pvx Z and numerical orbitals: 0.28 0.28 0.19 0.19 0.28 0.28 Π 0.19 Π 0.19 Trygve Helgaker (CTCC, University of Oslo) Basis sets for correlated calculations 12th Sostrup Summer School (2012) 22 / 24

Basis-set convergence of correlation energy electrons basis set Ne MP2 Ne CCSD N 2 MP2 N 2 CCSD H 2O MP2 H 2O CCSD valence 6-31G 113.4 114.3 236.4 225.8 127.8 134.4 6-31G 150.3 152.2 305.3 308.3 194.6 203.8 6-311G 209.0 210.6 326.4 326.3 217.4 224.9 cc-pvdz 185.5 189.0 306.3 309.3 201.6 211.2 cc-pvtz 264.3 266.3 373.7 371.9 261.5 267.4 cc-pvqz 293.6 294.7 398.8 393.1 282.8 286.0 cc-pv5z 306.2 305.5 409.1 400.6 291.5 292.4 cc-pv6z 311.8 309.9 413.8 403.7 295.2 294.9 extrapolated 319.5 315.9 420.3 408.0 300.3 298.3 R12 320(1) 316(1) 421(2) 408(2) 300(1) 298(1) all cc-pcvdz 228.3 232.2 382.7 387.8 241.3 251.8 cc-pcvtz 329.1 331.4 477.8 478.2 317.5 324.2 cc-pcvqz 361.5 362.7 510.7 507.1 342.6 346.5 cc-pcv5z 374.1 373.7 523.1 516.7 352.3 353.9 cc-pcv6z 379.8 378.2 528.7 520.6 356.4 356.9 extrapolated 387.6 384.4 536.4 526.0 362.0 361.0 R12 388(1) 384(1) 537(2) 526(2) 361(1) 361(2) Some observations: the 6-31G and G-31G** are much too small the correlation-consistent basis sets provide a smooth convergence as expected, convergence is slow, chemical accuracy is not reached even for cc-pv6z extrapolation is possible Trygve Helgaker (CTCC, University of Oslo) Basis sets for correlated calculations 12th Sostrup Summer School (2012) 23 / 24

Extrapolations Correlation-consistent basis sets are realizations of the principal expansion The error in the energy is equal to the contributions from all omitted shells: E X n=x +1 n 4 X 3 From two separate calculations with basis sets E X and E Y E =E X + AX 3 E =E Y + AY 3 we eliminate A to obtain the following two-point extrapolation formula: E = X 3 E X Y 3 E Y X 3 Y 3 Mean absolute error in the electronic energy of CH 2, H 2 O, HF, N 2, CO, Ne, and F 2 : me h DZ TZ QZ 5Z 6Z R12 plain 194.8 62.2 23.1 10.6 6.6 1.4 extr. 21.4 1.4 0.4 0.5 For the error in the AE of CO relative to R12, we now obtain: kj/mol DZ TZ QZ 5Z 6Z plain 73.5 28.3 11.4 6.0 3.5 extr. 18.5 0.7 0.0 0.0 Chemical accuracy is now achieved with just 168 AOs (QZ), at a fraction of the cost Trygve Helgaker (CTCC, University of Oslo) Basis-set extrapolation 12th Sostrup Summer School (2012) 24 / 24