Basis sets for electron correlation
Trygve Helgaker
Centre for Theoretical and Computational Chemistry
Department of Chemistry, University of Oslo, Norway
The 11th Sostrup Summer School
Quantum Chemistry and Molecular Properties
July 4–16 2010
Trygve Helgaker (CTCC, University of Oslo)
Basis setes for electron correlation
11th Sostrup Summer School (2010)
1 / 24
Introduction
I 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
I We have discussed basis functions and basis sets for uncorrelated methods
I we are now going to consider basis set for electron correlation
I
I
Overview
1
the Coulomb hole and Coulomb cusp
2
basis-set convergence of the correlation energy
I
I
I
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
11th Sostrup Summer School (2010)
2 / 24
The local kinetic energy
I Consider the local energy of the helium atom
Eloc = (HΨ)/Ψ
← constant for the exact wave function
I The electronic Hamiltonian has singularities at points of coalescence
1
2
2
1
1
−
+
H = − ∇21 − ∇22 −
2
2
r1
r2
r12
I
infinite potential terms canceled by infinite kinetic terms at coalescence
300
I Local kinetic energy in the helium atom
I
I
positive around the nucleus
negative around the second electron
200
100
I Negative kinetic energy counterintuitive
I
I
I
I
classical forbidden region
internal “tunneling”
w. f. decays towards the singularity
the Coulomb hole
0
!100
!0.5
0.0
0.5
0.5
1.0
Trygve Helgaker (CTCC, University of Oslo)
The Coulomb cusp and Coulomb hole
0.0
11th Sostrup Summer School (2010)
3 / 24
The Coulomb hole
I Each electron is surrounded by a classically forbidden region: the Coulomb hole
I
without a good description of this region, our results will be inaccurate
I The helium wave function with one electron fixed at a separation of 0.5a0 from the nucleus
I
total wave function with the corresponding Hartree–Fock wave function subtracted
0.5
0.0
- 0.5
0.00
- 0.05
- 0.10
- 1.0
- 0.5
0.0
0.5
1.0
Trygve Helgaker (CTCC, University of Oslo)
The Coulomb cusp and Coulomb hole
11th Sostrup Summer School (2010)
4 / 24
Cusp conditions
I Consider the helium Hamiltonian expressed in terms of r1 , r2 , and r12 :
H=−
2
1X
2 i=1
∂2
2 ∂
2Z
+
+
∂ri2
ri ∂ri
ri
!
−
∂2
2 ∂
1
+
−
2
∂r12
r12 ∂r12
r12
+ ···
I The nuclear cusp condition at ri = 0:
I
2Z
2 ∂
+
ri ∂ri
ri
Ψ
=0
⇒
∂Ψ = −Z Ψ (ri = 0)
∂ri ri =0
⇒
∂Ψ 1
= Ψ (r12 = 0)
∂r12 r12 =0
2
ri =0
easy to satisfy by the use of STOs
I The Coulomb cusp condition at r12 = 0:
I
1
2 ∂
−
r12 ∂r12
ri
Ψ
=0
r12 =0
impossible to satisfy by orbital-based wave functions
0.4
0.1
!1.0
Trygve Helgaker (CTCC, University of Oslo)
!0.5
0.5
The Coulomb cusp and Coulomb hole
1.0
11th Sostrup Summer School (2010)
5 / 24
Convergence of the helium ground-state energy
I The short-range interactions are difficult to describe
I
we must model the hole accurately for chemical accuracy in our calculations
I We shall compare the convergence of the following expansions for the helium ground-state
1
2
3
conventional CI
I single-zeta STOs
I numerical orbitals
CI with a correlating function
I CI-R12
the Hylleraas function
0.4
0.1
!1.0
Trygve Helgaker (CTCC, University of Oslo)
!0.5
0.5
Convergence of the helium ground-state energy
1.0
11th Sostrup Summer School (2010)
6 / 24
Configuration-interaction wave function for helium
I Our one-electron basis functions are STOs:
χn`m (r , θ, ϕ) = r n−1 exp (−ζr ) Y`m (θ, ϕ)
s
2` + 1 (` − m)! m
Y`m (θ, ϕ) =
P (cos θ) eimϕ
4π (` + m)! `
I
the associated Legendre polynomials P`m (x) are orthogonal on [−1, 1]
I The helium ground-state FCI wave function constructed from such STOs becomes:
ΨFCI (r1 , r2 ) = exp [−ζ (r1 + r2 )]
X
P`0 (cos θ12 )
`
I
X
r1n1 −1 r2n2 −1 + r2n1 −1 r1n2 −1
n1 n2
we have here used the addition theorem
P`0 (cos θ12 ) =
`
4π X
(−1)m Y`,m (θ1 , ϕ1 )Y`,−m (θ2 , ϕ2 )
2` + 1 −`
I Note: the CI expansion employs only three coordinates: r1 , r2 , cos θ12
I
the interelectronic distance r12 does not enter directly
Trygve Helgaker (CTCC, University of Oslo)
Convergence of the helium ground-state energy
11th Sostrup Summer School (2010)
7 / 24
The principal expansion
I Include in the FCI wave function all STOs up to a given principal quantum number:
N=1:
Ψ1 = |1s 2 |
N=2:
Ψ2 = c1 |1s 2 | + c2 |1s2s| + c3 |2s 2 | + c4 |2p 2 |
I The principal expansion converges very slowly
I
it is difficult to obtain an error smaller than 0.1 mEh
50
100
150
200
250
æ
à
10-2
æ
àæ
à æ
à
æ
à
-4
10
æ
à
æ
à
æ
à
æ
single-Ζ CI
æ
ànumerical CI
I The use of fully numerical orbitals reduces the error by a few factors
I
it does not improve on the intrinsically slow FCI convergence
Trygve Helgaker (CTCC, University of Oslo)
Convergence of the helium ground-state energy
11th Sostrup Summer School (2010)
8 / 24
Correlating functions
I By introducing
2 − r2 − r2
r12
1
2
2r1 r2
we may write the FCI wave function in the form
X
2k
ΨFCI (r1 , r2 , r12 ) = exp [−ζ (r1 + r2 )]
cijk r1i r2j + r2i r1j r12
cos θ12 =
ijk
I Since only even powers of r12 are included, the cusp condition can never be satisfied
∂ΨCI =0
∂r12 r12 =0
I However, if we include a term linear in r12
ΨCI
r12 =
1
1 + r12 ΨCI
2
then the cusp condition is satisfied exactly
∂ΨCI
1
1
r12 = ΨCI (r12 = 0) = ΨCI
(r12 = 0)
∂r12 2
2 r12
r12 =0
I We may always satisfy the cusp condition by multiplication with a correlating function:
γ =1+
Trygve Helgaker (CTCC, University of Oslo)
1
2
X
i>j
rij
Convergence of the helium ground-state energy
11th Sostrup Summer School (2010)
9 / 24
Explicitly correlated methods
I Methods that employ correlating functions or otherwise make explicit use of the
interelectronic distances rij are known as explicitly correlated methods
I
I
the R12 method includes rij linearly
the F12 method includes a more general (exponential) dependence on rij
I The R12 principal expansion
CI
CI
ΨR12
N = ΨN + c12 r12 Ψ1
converges easily to within 0.1 mEh (chemical accuracy)
50
100
150
200
250
æ
ì
æ
10-2 à
ìæ
ìæ æ
à ì
ì
10-4
à
à
à
æ
ì
à
æ
ì
à
single-Ζ CI
æ
æ
æ
ì
ìnumerical CI
à
à
10-6
single-Ζ CI-R12
à
I Still, it appears difficult to converge to within 1 µEh (spectroscopic accuracy)
Trygve Helgaker (CTCC, University of Oslo)
Convergence of the helium ground-state energy
11th Sostrup Summer School (2010)
10 / 24
The Hylleraas function
I Finally, we include in the wave function all powers of r12
ΨH (r1 , r2 , r12 ) = exp [−ζ (r1 + r2 )]
X
k
cijk r1i r2j + r2i r1j r12
ijk
I This wave function is usually expressed in term of the Hylleraas coordinates
t = r1 − r2 ,
s = r1 + r2 ,
u = r12
I Only even powers in t are needed for the singlet ground state:
ΨH (r1 , r2 , r12 ) = exp (−ζs)
X
cijk s i t 2j u k
ijk
I The Hylleraas function converges easily to within 0.1 µEh
50
100
150
200
250
æ
ì
ò
æ
ì
10-2 à
òæ
10-4
æ
àò ò
ìà
ìà
ì
10-6
10-8
æ
ò
æ
æ
single-Ζ CI
æ
ò
ò
æ
æ
ò
ònumerical CI
à
à
à
à
à
ì
ì
ì
ì
ì
ì
single-Ζ CI-R12
à
ì
Hylleraas
ì
I The Hylleraas method cannot easily be generalized to many-electron systems
Trygve Helgaker (CTCC, University of Oslo)
Convergence of the helium ground-state energy
11th Sostrup Summer School (2010)
11 / 24
Convergence rates
I We have seen the reason for the slow convergence of FCI wave functions
DZ
-90
90
QZ
TZ
-90
90
-90
5Z
90
-90
90
I Let us now examine the rate of convergence for the helium atom using the
1
2
partial-wave expansion
principal expansion
4f
1s
4f
5f
6f
3d
4d
3d 4d 5d 6d
2p
3p
4p
2p 3p 4p 5p 6p
2s
3s
4s
1s 2s 3s 4s 5s 6s
principal expansion
Trygve Helgaker (CTCC, University of Oslo)
partial-wave expansion
The partial-wave and principal expansions
11th Sostrup Summer School (2010)
12 / 24
The partial-wave expansion of helium
I Consider the expansion of the helium FCI wave function in partial waves:
ΨCI
L =
L
X
ψ`
`=0
I
this expansion has been studied in great detail theoretically
I Each partial wave is an infinite expansion in determinants
I
it contains all possible combinations of orbitals of angular momentum `, for example
1s 2 ,
1s2s,
2s 2 ,
1s3s,
3s3s,
3s 2 , . . .
I The contribution from each partial wave converges asymptotically as
EL = EL − EL−1 = −0.074226 L +
1 −4
2
− 0.030989 L +
1 −5
2
+ ···
I Convergence is slow but systematic
Trygve Helgaker (CTCC, University of Oslo)
The partial-wave and principal expansions
11th Sostrup Summer School (2010)
13 / 24
The principal expansion of helium
I The partial-wave expansion is difficult to realize in practice
I The alternative principal expansion contains a finite number of terms at each level
Ψ1 :
1s 2
Ψ2 : 1s 2 , 1s2s, 2s 2 , 2p 2
I The principal expansion is higher in energy at each truncation level (Eh ):
L
0
1
2
3
EL
−2.879
−2.901
−2.903
−2.904
N
1
2
3
4
EN
−2.862
−2.898
−2.902
−2.903
I However, the asymptotic convergence rate of the energy corrections is the same
EN = EN − EN−1 = c4 N −
Trygve Helgaker (CTCC, University of Oslo)
1 −4
2
The partial-wave and principal expansions
+ ···
11th Sostrup Summer School (2010)
14 / 24
Energy contributions and errors
I The contribution to the correlation energy from each AO in large helium CI calculations is
En`m = −an−6
X
⇒
4
En`m = − π90 a = −1.08a
n`m
I The contribution from each partial wave is therefore:
E` = −a (2` + 1)
∞
X
n−6 ≈ a (2` + 1)
Z
n−6 dn
`+1/2
n=`+1
= − 15 a(2` + 1) ` +
∞
1 −5
2
= − 52 a ` +
1 −4
2
I The asymptotic truncation error of the partial-wave expansion with ` ≤ L is therefore
∆EL = EL − E∞ = 52 a
∞
X
`+
1 −4
2
+ · · · ≈ 52 a
Z
`=L+1
∞
`+
L+1/2
1 −4
2
d` =
2
a (L
15
+ 1)−3
I The contribution from each shell in the principal expansion is:
En = −an2 n−6 = −an−4
I The asymptotic truncation error of principal expansion with n ≤ N is therefore
∆EN = EN − E∞ = a
∞
X
n=N+1
n−4 ≈ a
Z
∞
N+1/2
n−4 dn = 13 a(N + 21 )−3
I The two series converge slowly but smoothly and may therefore be extrapolated
Trygve Helgaker (CTCC, University of Oslo)
The partial-wave and principal expansions
11th Sostrup Summer School (2010)
15 / 24
Some observations
I The number of AOs at truncation level N in the principal expansion is given by
Nao =
N
X
n2 = 61 N(N + 1)(N + 2) ∝ N 3
i=1
I It follows that the error is inversely proportional to the number of AOs:
−1
∆EN ∝ N −3 ∝ Nao
I The dependence of the error in the correlation energy on the CPU time is thus:
∆EN ∝ T −1/4
I Each new digit in the energy therefore costs 10000 times more CPU time!
1 minute
→
1 week
→
200 years
I The convergence is exceedingly slow!
I A brute-force basis-set extension until convergence may not always be possible.
I Fortunately, the convergence is very smooth, allowing for extrapolation.
Trygve Helgaker (CTCC, University of Oslo)
The partial-wave and principal expansions
11th Sostrup Summer School (2010)
16 / 24
Basis sets for correlated calculations
I We must provide correlating orbitals for the virtual space
I The requirements are more severe than for uncorrelated calculations
I 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
11th Sostrup Summer School (2010)
17 / 24
Valence and core correlation
I The core electrons are least affected by chemical processes
I For many purposes, it is sufficient to correlate the valence electrons
I Example: the dissociation of BH
I
I
to the left, total electronic energies
to the right, core and valence correlation energies
corr
corr
corr
Ecore
= Eall
− Eval
-25.0
HF
core
-0.050
FCI HallL
valence
-25.2
FCI HfcL
1
2
3
4
-0.150
5
1
2
3
4
5
I The valence correlation energy can be recovered with smaller basis sets
Trygve Helgaker (CTCC, University of Oslo)
Basis sets for correlated calculations
11th Sostrup Summer School (2010)
18 / 24
Atomic natural orbitals (ANOs)
I ANOs are obtained by diagonalizing the one-electron CISD atomic density matrix
I We obtain a large primitive basis that is generally contracted
I 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
I The occupation numbers provide a natural criterion for selecting basis functions:
η1l
η2l
η3l
η4l
η5l
s
2.000000
1.924675
0.008356
0.000347
0.000021
Trygve Helgaker (CTCC, University of Oslo)
p
–
0.674781
0.004136
0.000331
0.000034
d
–
–
0.008834
0.000124
0.000016
Basis sets for correlated calculations
f
–
–
–
0.000186
0.000011
g
–
–
–
–
0.000018
11th Sostrup Summer School (2010)
19 / 24
Correlation-consistent basis sets
I The correlation-consistent basis sets constitute a realization of the principal expansion:
1
2
begin with a generally contracted set of atomic HF orbitals
add primitive energy-optimized correlating orbitals, one shell at a time
I The resulting correlation-consistent basis sets
cc-pVX Z,
X is the cardinal number
forms a hierarchical system:
SZ
2s1p
cc-pVDZ
+3s3p3d
3s2p1d
cc-pVTZ
+4s4p4d4f
4s3p2d1f
cc-pVQZ
+5s5p5d5f 5g
5s4p3d2f 1g
number of AOs
∝ X2
∝ X3
I The number of basis functions is given by
NX = 31 (X + 1)(X + 3/2)(X + 2)
I The proportion of the correlation energy recovered increases slowly:
X
%
2
67
3
88
4
95
5
97
6
98
I Extensions:
aug-cc-pVX Z,
Trygve Helgaker (CTCC, University of Oslo)
cc-pCVX Z,
aug-cc-pCVX Z
Basis sets for correlated calculations
11th Sostrup Summer School (2010)
20 / 24
cc-pVX Z basis sets
I cc-pVDZ: 3s2p1d
2
s
2
1
p
2
1
1
3
5
d
1
1
3
5
1
3
5
I cc-pVTZ: 4s3p2d1f
2
s
2
1
p
2
1
1
3
2
5
d
1
1
3
5
1
3
5
f
1
1
3
5
I cc-pVQZ: 5s4p3d2f1g
2
s
2
1
1
3
2
5
f
2
d
1
1
3
2
1
5
1
3
5
g
1
1
Trygve Helgaker (CTCC, University of Oslo)
p
1
3
5
1
3
5
Basis sets for correlated calculations
11th Sostrup Summer School (2010)
21 / 24
Correlation-consistent basis sets
I Percentage of correlation energy recovered with standard and numerical orbitals:
X
cc-pVDZ
numerical
2
77.1
85.6
3
93.0
95.6
4
97.3
98.0
5
98.7
98.9
I The Coulomb hole calculated with standard cc-pVX Z and numerical orbitals:
0.28
0.28
0.19
-Π
-А2
0.19
А2
Π
-Π
-А2
0.28
-А2
Trygve Helgaker (CTCC, University of Oslo)
Π
А2
Π
0.28
0.19
-Π
А2
0.19
А2
Π
-Π
-А2
Basis sets for correlated calculations
11th Sostrup Summer School (2010)
22 / 24
Basis-set convergence of correlation energy
electrons
valence
all
basis set
6-31G
6-31G∗∗
6-311G∗∗
cc-pVDZ
cc-pVTZ
cc-pVQZ
cc-pV5Z
cc-pV6Z
extrapolated
R12
cc-pCVDZ
cc-pCVTZ
cc-pCVQZ
cc-pCV5Z
cc-pCV6Z
extrapolated
R12
Ne MP2
−113.4
−150.3
−209.0
−185.5
−264.3
−293.6
−306.2
−311.8
−319.5
−320(1)
−228.3
−329.1
−361.5
−374.1
−379.8
−387.6
−388(1)
Ne CCSD
−114.3
−152.2
−210.6
−189.0
−266.3
−294.7
−305.5
−309.9
−315.9
−316(1)
−232.2
−331.4
−362.7
−373.7
−378.2
−384.4
−384(1)
N2 MP2
−236.4
−305.3
−326.4
−306.3
−373.7
−398.8
−409.1
−413.8
−420.3
−421(2)
−382.7
−477.8
−510.7
−523.1
−528.7
−536.4
−537(2)
N2 CCSD
−225.8
−308.3
−326.3
−309.3
−371.9
−393.1
−400.6
−403.7
−408.0
−408(2)
−387.8
−478.2
−507.1
−516.7
−520.6
−526.0
−526(2)
H2 O MP2
−127.8
−194.6
−217.4
−201.6
−261.5
−282.8
−291.5
−295.2
−300.3
−300(1)
−241.3
−317.5
−342.6
−352.3
−356.4
−362.0
−361(1)
H2 O CCSD
−134.4
−203.8
−224.9
−211.2
−267.4
−286.0
−292.4
−294.9
−298.3
−298(1)
−251.8
−324.2
−346.5
−353.9
−356.9
−361.0
−361(2)
I Some observations:
I
I
I
I
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
11th Sostrup Summer School (2010)
23 / 24
Extrapolations
I Correlation-consistent basis sets are realizations of the principal expansion
I The error in the energy is equal to the contributions from all omitted shells:
∆EX ≈
P∞
n=X +1
n−4 ≈ X −3
I From two separate calculations with basis sets EX and EY
E∞ =EX + AX −3
E∞ =EY + AY −3
we eliminate A to obtain the following two-point extrapolation formula:
E∞ =
X 3 EX − Y 3 EY
X3 − Y 3
I Mean absolute error in the electronic energy of CH2 , H2 O, HF, N2 , CO, Ne, and F2 :
mEh
plain
extr.
DZ
194.8
TZ
62.2
21.4
QZ
23.1
1.4
5Z
10.6
0.4
6Z
6.6
0.5
R12
1.4
I For the error in the AE of CO relative to R12, we now obtain:
kJ/mol
plain
extr.
DZ
−73.5
TZ
−28.3
−18.5
QZ
−11.4
−0.7
5Z
−6.0
0.0
6Z
−3.5
0.0
I 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
11th Sostrup Summer School (2010)
24 / 24