Electron Correlation: Single-Reference Methods

Jürgen Gauss I.

What is Electron Correlation? Correlation in Probability Theory

two variables x and y and probability densities P(x), P(y), P(x,y)

variables x and y independent if

P (x, y)=P (x)P (y)

otherwise

P (x, y) = P (x)P (y) 6

the variables are correlated Electron Correlation: the Helium Atom

nucleus r1 electrons helium atom: r12

r2

electron density ⇢(r)

pair density ⇢2(r1, r2) Electron Correlation: the Helium Atom

nucleus

electron 2

wave function ΨHF(r1,r2) as function of r1 Electron Correlation: the Helium Atom

nucleus

electron 2

HF description: independent motion Electron Correlation: the Helium Atom

nucleus r1 electrons helium atom: r12

r2

electron density ⇢(r)

pair density ⇢2(r1, r2) 1 HF pair density ⇢ (r , r )= ⇢(r ) ⇢(r ) 2 1 2 4 1 2 Electron Correlation: the Helium Atom

nucleus

electron 2

correlation effect

wave function Ψexact(r1,r2) as function of r1 Electron Correlation: the Helium Atom

nucleus

electron 2

correlation effect

correlated motion Electron Correlation: the Helium Atom

nucleus electron 2

Ψexact(r1,r2) - ΨHF(r1,r2) , as function of r1 Electron Correlation: the Helium Atom

nucleus electron 2

Coulomb hole Electron Correlation: the Helium Atom

nucleus r1 electrons helium atom: r12

r2

electron density ⇢(r)

pair density ⇢2(r1, r2) 1 exact pair density ⇢ (r , r ) = ⇢(r ) ⇢(r ) 2 1 2 6 4 1 2 Electron Correlation: the Extreme Case

two interacting electrons two possible sites

probabilities:

one electron at 50 %

one electron at 50 % Electron Correlation: the Extreme Case

two interacting electrons two possible sites mean-field (HF) description not favorable!

both electrons at 25 %

both electrons at 25 %

one electron at and one at 50 % Electron Correlation: the Extreme Case

two interacting electrons two possible sites

avoids unfavorable correlated description combinations

both electrons at 0 %

both electrons at 0 %

one electron at and one at 100 % Different Types of Electron Correlation

• Fermi correlation (exchange)

electrons of same spin avoid each other due to the antisymmetry requirement (à Fermi hole)

already treated at HF level

independent of Coulomb interactions

• Coulomb correlation electrons (independent of spin) avoid each other due to the repulsive Coulomb interactions (à Coulomb hole)

not treated at HF level, requires post-HF methods Electron-Correlation Energy

usual definition (Löwdin, 1955):

E = E E corr nrl HF

• focus on Coulomb correlation

• Fermi correlation in EHF already included

• HF for electron correlation reference point Different Types of Electron Correlation

• dynamical correlation

electrons (independent of spin) avoid each other due to the repulsive Coulomb interactions (à Coulomb hole) • static (non-dynamical) correlation

(quasi-)degeneracy of Slater determinants/configurations

1 2 E

E( ) E( ) à c11 + c22 1 ⇡ 2 ⇡ Single- and Multireference Methods no static correlation

HF quality correct à good starting point for correlation treatment

one Slater determinants dominates

applies for many/most molecules in and close to their equilibrium configuration

à single-reference treatment static correlation

multi-reference treatment required II.

Importance of Electron Correlation Magnitude of Correlation Energies

H2O HCN

EHF -76.068 -92.916

ΔEcorr -0.372 -0.518

ΔErel -0.052 -0.044

correlation energies typically < 1% of the total energies Chemical Relevance of Electron Correlation

0.001 Hartree ˆ= 2.6255 kJ/mol

correlation energies strongly dependent on valence electrons correlation effects always important when bonds are broken Chemical Relevance of Electron Correlation

example: CO à C(3P) + O(3P)

dissocation energy (De) of CO

C O CO De in a.u. E(HF) -37.693774 -74.819232 -112.790997 .277991 E(corr) -.151537 -.248978 -.536591 .136076 E(total) -37.845307 -75.068210 -113.327588 .414071 Chemical Relevance of Electron Correlation

example: CO à C(3P) + O(3P)

dissocation energy (De) of CO

C O CO De in kJ/mol E(HF) -37.693774 -74.819232 -112.790997 729.9 E(corr) -.151537 -.248978 -.536591 357.3 E(total) -37.845307 -75.068210 -113.327588 1087.2

electron-correlation effects are significant III.

Perturbative Treatment of Electron Correlation:

Møller-Plesset Perturbation Theory Hamilton Operator for Many-Electron Systems

non-relativistic Hamiltonian for atoms and molecules

1 Hˆ = hˆ()+ r ⇥ <⇥

one-electron terms two-electron terms prohibits exact solution à electron correlation Møller-Plesset Perturbation Theory

electron correlation small effect

à treatment via perturbation theory on top of HF perturbation theory:

(0) zeroth-order à HF theory = HF Møller-Plesset ˆ (0) ˆ (0) H = F (↵) E = "i ansatz ↵ i X X E = Hˆ HF h HF | | HF i (0) electron correlation = Hˆ + Hˆ 0 h HF | | HF i in second order = E(0) + E(1) MP2 Perturbation Theory

second-order perturbation theory

(0) ˆ (0) (0) ˆ (0) (2) 0 H I I H 0 E = h | | (0) ih (0)| | i I=0 E0 EI X6 other HF wavefunction eigenfunctions to H(0)

all Slater determinants constructed from (0) eigenfunctions to H : occupied and virtual canonical HF orbitals Excitation Level of Slater Determinants

virtual ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ occupied occupied

singly doubly triply ... HF excited excited excited Slater-Condon Rules and Brillouin Theorem

Slater-Condon rules rules for matrix elements involving two different Slater determinants

for H containing one- and two-electron operators

Hˆ =0 h I | | J i6

only if ΦI and ΦJ differ at most by a double excitation

Brillouin theorem

Hˆ a =0 h HF | | i i singly excited determinant MP2 Perturbation Theory

second-order perturbation theory

(0) ˆ (0) (0) ˆ (0) (2) 0 H I I H 0 E = h | | (0) ih (0)| | i I=0 E0 EI X6

leads to antisymmetrized ab ij 2 two-electron integral E = |h || i| MP2 " + " " " i

increment total ΔE(MP2) -0.2040 -0.2040 ΔE(MP3) -0.0068 -0.2108 ΔE(MP4) -0.0052 -0.2160

ΔEcorr -0.2170 -0.2170

H2O, cc-pVDZ basis

MP2 recovers typically more than 95 % of electron correlation

MP3 involves only doubles, singles and triples only at MP4 IV.

Exact Solution of the Correlation Problem:

Full Configuration Interaction Exact Solution of Electron-Correlation Problem

define effective (zeroth-order) Hamiltonian

Hˆ0 = Fˆ() solve one-electron problem (e.g., HF equations) Fˆ ⇥p = p ⇥p Slater determinants (take care of antisymmetry) 1 „complete“ one-electron basis I = I1 I2 ... IN N! | | (spin orbitals)

p,p=1, ... ,I =1, ... { } { I } complete many-electron basis within given AO basis Exact Solution of Electron-Correlation Problem

expansion in a complete set of Slater determinants

⇥ = cI I = c0 HF + cI I I I=0 excited determinants together with variational principle

H c = Ec H = Hˆ IJ J I IJ I | | J ⇥ J ⇒ Full Configuration Interaction (FCI) Excitation Level of Slater Determinants

virtual ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ occupied occupied doubly triply singly ... HF excited excited excited

2 2 3 3 number 1 noccNvirt nocc Nvirt nocc Nvirt

Full CI Approach

Full CI: • factorial growth of cost • not practial, only for benchmarking

small molecules and basis (e.g., H2O, DZP) benzene, DZP: ≈ 1042 determinants typical examples: E(FCI), in a.u.

BH, TZP+ -25.243140

H2O, cc-pVDZ -76.243773 CO, cc-pVDZ -113.055853 V.

Truncated Configuration-Interaction Schemes Full and Truncated CI Methods

Full CI: • factorial growth of cost • not practial, only for benchmarking

=> truncate determinantal basis (approximation) Excited Slater Determinants

virtual ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ occupied occupied

singly doubly triply ... HF excited excited excited

2 2 3 3 number 1 noccNvirt nocc Nvirt nocc Nvirt

Truncated CI Methods

truncate determinantal basis (approximation)

chose determinants according to excitation level

double excitations are most important

=> truncated CI methods such as CISD

⇥CISD = c0 0 + cI I I S,D excited determinants restricted to single + double excitations Solution of the CI Problem

CI problem: Hc = E c

eigenvalue problem for matrix H

no direct diagonalization of H (too expensive)

Davidson scheme:

• diagonalize H in a (small) iterative subspace • expand subspace until convergence

• computational cost due to product H c · Convergence to FCI

example: H2O, cc-pVDZ

CI method Energy [a.u.] CISD -76.231972 CISDT -76.234926 CISDTQ -76.243457 CI(5) -76.243640 CI(6) -76.243770 CI(7) -76.243772 CI(8) -76.243773 CI(9) -76.243773 FCI -76.243773 Computational Cost

approximation cost CISD N6 CISDT N8 CISDTQ N10 CI(5) N12 … …

FCI VI.

Size Consistency and Size Extensivity Separability for Non-Interacting Systems non-interacting systems A and B

A B

rAB → ∞

EAB = EA + EB

• additivity of energy • multiplicative separability of wavefunction Size Consistency

A B

rAB → ∞

A method is termed size consistent if the sum of energies computed for two non-interacting subsystems A and B is equal to the energy obtained for the supersystem consisting of both A and B Pople, Binkley, Seeger (1976)

a) individual quantum-chemical calculations for A and B à EA + EB b) one quantum-chemical calculation for A+B à EAB

size consistency: EAB = EA + EB Size Extensivity

... A ... A ... A ... A ... => E = n E · A

A size-extensive method provides results that scale linearly with the size of the system Bartlett, Purvis (1978)

- formal definition, independent of non-interacting reference systems - exploits full power in the context of diagrammatic techniques - size extensivity implies size consistency, the reverse is not true - ensures quality of results independent of size of systems Size-Consistency Problem of Truncated CI

CISD approximation:

2 x EHF EHF···HF Δ HF-SCF -200.558 -200.558 0.000 HF ··· HF CISD -200.576 -200.559 -0.017 ≈11 kcal/mol

calculations with tzp basis set

truncated CI methods are not size consistent Empirical Size-Consistency Corrections

Davidson correction:

to be added to the CISD energy, approximately size consistent

Coupled-Pair Functional Methods:

restore approximate size consistency by modifying the denominator in the energy expression to be minimized

CPF, ACPF, AQCC, ... due to Ahlrichs, Gdanitz, Szalay, Bartlett, ... VII.

Exponential Ansatz for Wavefunction Multiplicative Separability of the Wavefunction

non-interacting systems A and B

HˆAB = HˆA + HˆB

=> separation ansatz

EAB = EA + EB additive

AB = ˆ A B multiplicative A

antisymmetrizer

Multiplicative Separability of the Wavefunction

↑

H2 molecule, minimal basis ↑ ↑

↑

CID = HF + c D HF D (not normalized) HF double determinant excitation = (1 + ˆD) HF

operator, generates a by c weighted double excitations

H2: CID = FCI = exact solution Multiplicative Separability of the Wavefunction

two H2 molecules, minimal basis H2(A) + H2(B) CID wavefunction

CID = (1 + ˆD,A + ˆD,B) HF but

ˆ = ˆ (1 + ˆ ) (1 + ˆ ) A CID,A CID,B A{ D,A HF,A}{ D,B HF,B} = ˆ (1 + ˆ ) (1 + ˆ ) A D,A D,B HF,A HF,B = (1 + ˆD,A) (1 + ˆD,B) HF

= (1 + ˆD,A + ˆD,B + ˆD,A ˆD,B) HF

quadruple excitation, missing in CID => problem Multiplicative Separability of the Wavefunction

problem of CI:

excitations are additive

ˆD,i i=A,B,... contradiction !?! required is a product

... i=A,B,... Multiplicative Separability of the Wavefunction

one, two, three, ... H2 molecules

1 = (1 + ˆD,A) HF

2 = (1 + ˆD,A) (1 + ˆD,B) HF

3 = (1 + ˆD,A) (1 + ˆD,B) (1 + ˆD,C ) HF

...

= (1 + ˆD,i) HF i=A,B,... Exponential Form for the Wavefunction proper form of wavefunction

= (1 + ˆD,i) HF i=A,B,... rewrite in exponential form

(1 + ˆ ) exp( ˆ ) D,i HF D,i HF i=A,B,... i=A,B,...

multiplicative separability via additivity in exponent Exponential of Operators

definition via a power series

1 1 1 exp(Aˆ)=1+Aˆ + Aˆ2 + Aˆ3 + ... + Aˆn + ... 2! 3! n!

exponential of a number à non-terminating power series exponential of an operator à power series can terminate depending on operator Exponential Form for the Wavefunction

(1 + ˆ ) exp( ˆ ) D,i HF D,i HF i=A,B,... i=A,B,... proper counting

2 1 1 1 ˆ ˆ ˆ + ˆ ˆ + ... 2 D,i 2 D,A D,B 2 D,B D,A i ˆ ˆ + ... D,A D,B same excitation cannot ˆ ˆ =0 be applied twice D,A D,A Exponential Form for the Wavefunction

exponential ansatz

= exp(Tˆ) 0

excitations reference determinant represented via an operator

proper multiplicative behaviour => size extensivity Historical Remarks

• statistical physics - power expansion of partition function for interacting gases Ursell (1927), Mayer (1941) • nuclear physics - first use of exponential ansatz in quantum-mechanical context - quantum-mechanical description of nuclear matter F. Coester, H. Kümmel (1958, 1960) • electronic-structure theory - application to the electron-correlation problem => coupled-cluster theory J. Čížek (1966)

- see also earlier work by Hubbard (1957) and Sinanoğlu (1962) Connected and Disconnected Excitations

= (1 + ˆD,i) HF i=A,B,...

= HF + ˆD,A HF + ... + ˆD,AˆD,B HF + ...

connected excitations disconnected excitations (products of connected excitations)

CI does not differ between connected and disconnected excitations VIII.

Coupled-Cluster Theory Coupled-Cluster Ansatz for Wavefunction ansatz for wavefunction

= exp(T ) 0 | | reference cluster operator determinant

T = tI I excitation operators I (in second quantization)

weighting factors sum over unknown parameters (amplitudes) to be determined (all) excitations Cluster Operator

classification of excitation T = T1 + T2 + T3 + ...

a T1 = ti aa† ai single excitations a i 1 ab T = t a† a a†a double excitations 2 4 ij a i b j i,j a,b ... all possible excitations T = T1 + T2 + T3 + ... + TN

truncated schemes, e.g. T = T1 + T2 Equivalence of CC and FCI parameterization

CC wavefunction CI wavefunction = exp(T ) 0 = C 0 | | | | T 0 singly excited C 0 1 | 1 | 1 2 (T + T ) 0 doubly excited C2 0 2 2! 1 | | 1 3 (T3 + T2T1 + T1 ) 0 triply excited C 0 3! | 3 | 1 2 (T4 + T3T1 + T2 quadruply excited C4 0 1 2! 1 | + T 2T + T 4) 0 2! 1 2 4! 1 | ... matching number of parameters

=> CC ansatz parameterizes the exact wavefunction Truncated Coupled-Cluster Wavefunctions

T = T2 truncation in the cluster operator

CCD = CC doubles

1 1 = 0 + T 0 + T 2 0 + T 3 0 + ... | CCD | 2 | 2! 2 | 3! 2 |

connected disconnected double excitations quadruple, sextuple, ... excitations Coupled-Cluster Ansatz

• ansatz for many-electron wavefunction

- able to parameterize exact solution

- multiplicatively separable à size consistency and extensivity

- unknown parameters: amplitudes in cluster operator

• determination of actual CC wavefunction

- determine amplitudes in cluster operator - solve Schrödinger equation with CC ansatz what is a suitable procedure and what to do in the case of approximations? Variational Coupled-Cluster Theory

determination of amplitudes via variation principle

0 exp(T †)H exp(T ) 0 minimization of E˜ = | | 0 exp(T ) exp(T ) 0 | † | CC wavefunction not normalized adjoint of CC wavefunction

non-truncating operator

1 2 1 2 exp(T †)H exp(T )=H + T †H + HT + T † H + T †HT + HT + ... 2! 2! cost independent of choice of T always similar to FCI => not feasible for practical schemes

see work by Van Voorhis, Head-Gordon (2000), recent work on quasi-variational CC by Knowles and co-workers (2010), and work on XCC by Bartlett and Noga (1988) Algebraic Equations via Projection

transform an operator equation into a set of algebraic equations

H = E | |

I ,I =1,... equivalent { complete basis }

H = E I =1,... I | | I | matching number of equations and wavefunction parameters (amplitudes) Coupled-Cluster Theory

• insertion into Schrödinger equation H exp(T ) 0 = E exp(T ) 0 | | • multiplication from the left with exp(-T)

exp( T ) H exp(T ) 0 = E 0 | | • projection onto reference determinant

→ CC energy E = 0 exp( T ) H exp(T ) 0 | |

• projection onto excited determinants

→ CC equations 0= P exp( T ) H exp(T ) 0 | | non-linear equations for amplitudes Standard CC Theory

• untruncated CC approaches matching number of equations and parameters (amplitudes) - projection on all excited determinants - same number of excitations as in many-electron basis

• truncated CC approaches - projection on all excited determinants à unsolvable problem - project only on those determinants for which excitations are in T

T = T 2 à project onto D

T = T 1 + T 2 à project onto S, D

Schrödinger equation can no longer be solved in an exact manner Incomplete Projection

solving CC equations 0= exp( T ) H exp(T ) 0 P | | analysis of matrix H¯ = exp( T ) H exp(T ) PQ P | | Q full CC truncated CC

ECC ECC 0 0 0 ··· 0 ··· 0 0 all entries zero block not zero !!

=> equivalent to FCI => approximate solution Coupled-Cluster Equations

E = 0 exp( T ) H exp(T ) 0 | |

0= exp( T ) H exp(T ) 0 P =1, ... P | |

How do we continue from here? Coupled-Cluster Equations

„Simplifications“

Baker-Campbell-Hausdorff expansion

1 exp( T ) H exp(T )=H +[H, T]+ [[H, T],T]+... 2!

expansion in terms of commutators Normal Ordering

in the normal-ordered product form of an operator, N(A), all annihilation operators are to the right of the creation operators

N(ˆa aˆ†aˆ )= aˆ†aˆ aˆ p q r q p r

sign: (-1)p with p as number of required transpositions to reach normal ordering advantage of normal ordering

VAC N(...aˆ† aˆ ...) VAC =0 except N(…) = number | p q | ⇥

expectation values of normal-ordered operators are simple to evaluate! Normal Ordering via Commutators

aˆ aˆ aˆ†aˆ† =ˆa [ˆa , aˆ†] aˆ† aˆ aˆ†aˆ aˆ† p q r s p q r + s p r q s

= [ˆa , aˆ†] aˆ†aˆ +[ˆa , aˆ†] aˆ aˆ† +ˆa†aˆ aˆ aˆ† qr p s + qr s p p r + q s r p q s

. . .

= aˆ†aˆ + aˆ†aˆ + qr ps pr qs qr s p pr s q

aˆ†aˆ aˆ†aˆ +ˆa†aˆ†aˆ aˆ qs r p ps r q r s p q Contractions contraction of a pair of operators

general definition

specific cases

≠ 0 only, if operators are not in normal order Wick’s Theorem an arbitrary product of creation and annihilation operators

ˆ X = ... aˆp† aˆqaˆr†aˆs ...

is equal to the normal-ordered product, N(X), plus all normal-ordered products of X that contain one, two, three, … contractions of two operators Wick’s Theorem: Example

aˆpaˆqaˆr†aˆs† = N(aˆpaˆqaˆr†aˆs†) = aˆr†aˆs†aˆpaˆq

+N( aˆpaˆqaˆr†aˆs†) +0

+N( aˆpaˆqaˆr†aˆs†) + praˆs†aˆq

+N( aˆ aˆ aˆ aˆ ) aˆ†aˆ p q r† s† ps r q +N(aˆ aˆ aˆ aˆ†) aˆ†aˆ p q r† s qr s p

+N(ˆap aˆqaˆr†aˆs†) + qsaˆr†aˆp

+N(aˆpaˆq aˆr†aˆs†) +0

+N( aˆpaˆq aˆr†aˆs†) +0

+N( aˆp aˆqaˆr†aˆs†) +psqr

+N( aˆ aˆ aˆ aˆ ) qspr p q r† s†

AˆBˆ = N(AˆBˆ)+ N( AˆBˆ )+ N( AˆBˆ )+... Quasi-Particle Formalism

as reference for VAC | N-electron systems not useful

useful for N-electron systems, HF 0 as only changes wrt | ⇥| ⇥ HF reference are considered

=> particle-hole formalism Quasi-Particle Formalism

HF state aˆ† 0 aˆ 0 a | i | virtual particle particles ↑ ↑ ↑ hole ↑ ↑ ↑ ↑ ↑ ↑ holes occupied occupied

Fermi vacuum particle creation hole creation Quasi-Particle Formalism

new definition of creation and annihilation operators

hole creation ˆ ˆ bi† :=a ˆi bi :=a ˆi† and annihilation

ˆ ˆ ba† :=a ˆa† ba :=a ˆa particle creation and annihilation

unchanged anti-commutator relations

usual convention: indices i,j,k, … occupied; a,b,c, … virtual; p,q,r, … generic 0 ... aˆ aˆ+aˆ ... 0 ⇤ | p q r | ⌅

0 ... aˆ aˆ+aˆ ... 0 Hamilton⇤ | p q Operatorr | ⌅ in Normal-Ordered Form + + aˆa aˆi aˆj = aˆj aˆa aˆi { 0 ...} aˆ aˆ+aˆ ... 0 ⇤ | p q r | ⌅ + + aˆa aˆi aˆj = aˆj aˆa aˆi 1 { } 1 1 H = EHF + + fpq1 p†q + + + pq rs p†q†sr H = E + f p q + { }pq rs4 p q srij kl|| i†⇥j{†lk } ij ij HF + pq p,q + 4p,q,r,s⌅ || ⇧{ } ⇤ 2 ⌅ || ⇧ aˆa aˆ aˆj{ = }aˆj aˆa 4aˆ ⇤ || ⌅{ } i,j { p,q i } ip,q,r,s i,j,k,l + 1 + + H = EHF + fpq p 1q + pq rs p q sr 1 { } 4 ⇤ || ⌅{ } p,q ijp,q,r,skl i† j†1lk ij ij 1 4 ⌅ || ⇧{ } ⇤ij kl1 2 i j lk⌅ || ⇧ ij ij i,j,k,l 1 †i,j † hij i†j + hii 4 ⌅4 || +⇧{+pi qj} p⇤† i†qj 2 ⌅ || ⇧ pi qi H = EHF{ +} ⇥fpq p q + i,j,k,lpq rs⇥p4 q sr⌅ || ⇧{ } i,j⇤ ⌅ || ⇧ i,j { i} 4 ⇤|| ⌅{ p,q } p,q p,q p,q,r,s hij i†j hii { } ⇥ i,j 1i 4 pi qj p† i†qj1 pi qi p†q 1 1 ⇥4 ⌅1 ||1 ⇧{ } ⇤ ⌅ || ⇧ { } p,q 4 pi qj p,qp† i†qj pi qi p†q ij klij klih†ijj†lkii††jj†lk hii ij ij⇥ij4ij ⌅ || f⇧{ = h } + ⇤ pi qi⌅ || ⇧ { } 4 4 ⇤ || ⌅ ⌅{|| ⇧{{ }} }⇥⇤ 2 2 ⇤ ⌅|| ||⌅ ⇧p,q pq pq ⌅ p,q|| ⇧ 1 i,j,k,l i,j,k,l i,j 1 i i,j i,j i ijkl i† j†lk ijij 4 ⇤ || ⌅{ } ⇥ 2 ⇤ || ⌅ Fock matrix i,j,k,l i,j f = h + pi qi 1 1 1 pq1 1 pq ij klij kli j lki† j†lk ij ijij ij ⌅fpq||=⇧ hpq + pi qi 44 ⇤ ||pi† ⌅{qj† p† i†}qj ⇥ 2 ⇤ pi|| iqi⌅ p†q ⌅ || ⇧ 14 ⇥⇤i,j,k,l4|| ⌅{⌅ || ⇧{} ⇥}1 ⇤2 i,j⇤ ||⌅ ⌅|| ⇧ { } i i,j,k,l ij klp,q i† j†lk i,jij ijp,q 4 ⇤ || ⌅{ } ⇥ 2 ⇤ ||⌅ i,j i,j,k,l 1 1 ij kl i† j†lk ij ij 4 ⇤ f|| =⌅{ h + } ⇥pi qi2 ⇤ || ⌅ i,j,k,l pq pq ⌅ || ⇧ i,j i Hamilton Operator in Normal-Ordered Form

H = EHF

+ f i†j + f i†a ij{ } ia{ } i,j i,a + f a†i + f a†b ai{ } ab{ } a,i a,b 1 1 + ij kl i†j†lk + ij ka i†j†ak 4 || ⇥{ } 2 || ⇥{ } i,j,k,l i,j,k,a 1 1 + ia jk i†a†kj + ij ab i†j†ba 2 || ⇥{ } 4 || ⇥{ } i,j,k,a i,j,a,b 1 + ab ij a†b†ji + ai bj a†i†jb 4 || ⇥{ } || ⇥{ } i,j,a,b i,j,a,b 1 1 + ai bc a†i†cb + ab ci a†b†ic 2 || ⇥{ } 2 || ⇥{ } i,a,b,c i,a,b,c 1 + ab cd a†b†dc 4 || ⇥{ } a,b,c,d Hamilton Operator in Normal-Ordered Form

H = EHF + fN + WN

two-body HF energy one-body operator operator

H = H 0 H 0 N ⇥| | ⇤

= fN + WN Coupled-Cluster Equations

„Simplifications“

• normal-ordered representation of commutators

[A, B]=AB BA = AB BA + AB BA { } { } { } { }

=0 Coupled-Cluster Equations

„Simplifications“

• T only consists of quasi-particle creation operators

TH =0, etc. { } [H, T]= HT + HT + ... { } { } (HT) c 1 exp( T ) H exp(T )=H +(HT) + ((HT) T ) + ... c 2! c c

(H exp(T )) c CCD Equations

E = 0 (H exp(T )) 0 | c |

0= (H exp(T )) 0 P | c | with

H = EHF + fN + WN T = T2

normal-ordered Hamiltonian only double excitations

and projection on doubly excited determinants ab D ij Excitation Rank count excitation levels

- projection on ΦP defines overall excitation levels

- Tn corresponds to an n-tuple excitation

- fN corresponds to excitation by -1, 0 , 1

- WN corresponds to excitation by -2, -1, 0 , 1, 2

example

WN T2T2 à corresponds to 2+2+(-2,-1,0,1,2) à (2,3,4,5,6) excitations

no contribution to projection on ΦS, but for those on ΦD, ΦT, ... CCD Approximation

H = EHF + fN + WN T = T2

CCD energy

E = 0 (H exp(T )) 0 | c | 1 => E = 0 H + (HT ) + ((HT ) T ) + ... 0 | 2 c 2! 2 c 2 c |

=> E = E + 0 (W T ) 0 HF | N 2 c| CCD Approximation

H = EHF + fN + WN T = T2

projection on doubly amplitude equations excited determinants

0= (H exp(T )) 0 D| c | 1 => 0= H + (HT ) + ((HT ) T ) + 0 D| 2 c 2! 2 c 2 c ··· |

=> 0= W + (f T ) + (W T ) D| N N 2 c N 2 c 1 + ((W T ) )T ) 0 2! N 2 c 2 c| Connectedness consider connectedness of terms

at most two connections for fN

at most four connections for WN possible excitation

fN: no connection (-1,0,1) one connection (-1,0) two connections (-1)

WN no connection (-2,-1,0,1,2) one connection (-2,-1,0,1) two connections (-2-1,0) three connections (-2,-1) four connections (-2) CCSD Equations

H = EHF + fN + WN T = T1 + T2 1 E = E + 0 (f T ) + (W T ) + ((W T ) )T ) 0 HF | N 1 c N 2 c 2! N 1 c 1 c |

projection on singly excited determinants

0= fN S| ++ (fN T1)c ++ (WN T1)c ++ (fN T2)c ++ (WN T2)c

1 1 ++ ((f T ) T ) ++ ((W T ) T ) 2! N 1 c 1 c 2! N 1 c 1 c singly excited 1 + ((WN T2)cT1)c + (((WN T1)cT1)c)T1)c 0 determinants 3! | CCSD Equations

H = EHF + fN + WN T = T1 + T2

projection on doubly excited determinants

0= D W ++ (WN T1)c ++ (fN T2)c ++ (WN T2)c | N 1 + ((W T ) T ) ++ ((f T ) T ) ++ ((W T ) T ) 2! N 1 c 1 c N 2 c 1 c N 2 c 1 c 1 + ((W T ) T ) 2! N 2 c 2 c 1 1 doubly excited + (((WN T2)cT1)c)T1)c + (((WN T1)cT1)c)T1)c determinants 2! 3! 1 + ((((WN T1)cT1)c)T1)c)T1)c 0 4! |

fN T1 and (((fN T1)cT1)cT1)c cannot contribute due to the connectness requirement Why do the CC Equations Terminate?

each T must be „connected“ to H

H only contains up to two-body operators

à at most 4 quasi-particle annihilation operators available for contractions

à at most 4 T operators can be „connected“ with H

=> (H exp(T))c contains at most quartic terms! Size Extensivity

CC equations contain only „connected“ terms

correct (linear) scaling with => size extensivity size of systems/number of electrons

ensures that quality of results is independent of size

connected terms linear scaling

disconnected terms quadratic, cubic, ... (unphysical) scaling Size Extensivity

disconnected term (WT)d connected term (WT)c system A one contribution one contribution (W from A, T from A) (W from A, T from A) system A+B

four contribution two contribution (W from A, T from A (W from A, T from A W from A, T from B, W from B, T from B) W from B, T from A W from B. T from B) quadratic scaling (wrong!) linear scaling Detailed Expressions for the CC Equations

• algebraic analysis via Wick’s theorem

- tedious, not recommended - suitable for computer algebra

• analysis by means of diagrammatic techniques

- simple access via graphical representations - automatic elimination of non-contributing term

- (better) suitable for computer algebra IX.

Diagrammatic Techniques in CC Theory Diagrammatic Representation: Operators

operators f a† a f pq{ p q} N

pq rs a† a†a a W || { p q s r} N ab t a† a†a a T ij { a b j i} product of creation and matrix element annihilation operators graphical representation f N T symbol with vertices x . number of vertices = WN number of p+q pairs ...... Diagrammatic Representation: Operators

QP-creation/annihilation operators lines QP-creation operators (QPC) lines above vertex QP-annihilation operators (QPA) lines below vertex

X . X . X .

2 QPC 2 QPA 1 QPC and QPA arrow direction

particles (virtual orbitals) upwards

holes (occupied orbitals) downwards Diagrammatic Representation: Operators

at each vertex there is one incoming and one outgoing line (conservation of particle number)

a† a† aa† ai a b (non existing)

X . X .

correct wrong Diagrammatic Representation of the Hamilton Operator

fai aa† ai f a†a fab a† ab f a†a { } ij{ i j} { a } ia{ i a}

† ab ij aa† abajai ia jk a†a† a a ab cd a† a†adac || { } || { i a k j} || { a b } ab ci a† a†aiac ai bj a† a†a a || { a b } || { a i j b}

ij ka a†a†a a ij kl a†a†a a i j a k || { i j l k} || { } ai bc a† a†a a ij ab ai†aj†abaa || { a i c b} || { } Diagrammatic Representation of the Cluster Operator

etc.

a ab abc t a† a t a† a†ajai t a† a†a†akajai i { a i} ij { a b } ijk { a b c } Diagrammatic Representation: Contractions product of two operators A and B

A

A B

B

left operator (A) above right operator (B) Diagrammatic Representation: Contractions

normal-ordered product form according to Wick’s theorem

all „diagrams“, in which the two operators are connected via none, one, two, … lines

connection = contraction connection only possible in the case of same directions of arrows

connections only between QPC in B and QPA in A

Diagrammatic Representation: Contractions example open lines (remaining QPC and QPA)

A A

AB internal line

B B

no contraction one contraction Diagrammatic Representation: Contractions example a (f T ) 0 i | N 1 c| 2 open lines

+ Connected, Disconnected, Linked, Unlinked

open with open lines closed no open lines connected all parts of the diagram are connected via lines

not all parts of the diagram disconnected are connected via lines linked closed and „connected“ unlinked closed and „disconnected“ Diagrammatic Representation of the CCD Model

CC energy:

CC equations:

Algebraic Analysis of Diagrams

Each line of the diagram corresponds to an index. The indices of the open lines correspond to the „target“indices (i, j, k, … in the case of holes and a, b, c, … in the case of particles), the indices of the internal lines correspond to the „summation“ indices (m, n, o, … in the case of holes and e, f, g, …in the case of particles)

a i b j target indices f e m n summation indices Algebraic Analysis of Diagrams

The symbols for the operators are replaced by the corresponding matrix elements.

out f(out, in) in out2 in1 in2 out1 out1 out2 in1 in2 || ⇥ out2 in1 in2 out1 out1 out2 tin1 in2 Algebraic Analysis of Diagrams

sum over all indices of the internal lines

a i b j target indices f e m n summation indices

=> tea tfb mn ef mn ij || m,n e,f Algebraic Analysis of Diagrams

n equivalent lines (same arrow direction, connection to same operator symbols) yield a factor of 1/n!

a i b j target indices f e m n summation indices

equivalent lines → factor 1/2

1 tea tfb mn ef => 2 mn ij || m,n e,f Algebraic Analysis of Diagrams

n equivalent operators (connected to the same operator(s) via lines of the same arrow direction) yield a factor of 1/n!

equivalent T2 operators => factor 1/2 Algebraic Analysis of Diagrams the sign of a diagram is given as (-1)h+l with h as the number of „internal hole lines“ and l as the number of „loops“

a i b j f e m n

loop internal hole lines

=> negative sign Algebraic Analysis of Diagrams sum over all non-equivalent permutations of the indices of the open hole and particle lines, respectively non equivalent equivalent a i b j f e m n

1 ea fb P (ab) tmn tij mn ef 2 || m,n e,f sign: (-1)p with p as the parity of permutation P (ab) Z(a, b)=Z(a, b) Z(b, a) Diagrammatic Representation of the CCD Model

CC energy:

CC equations:

CCD: Algebraic Expressions

CC energy:

CC equations: X.

Computational Realization of CC Approaches Solution of CC Equations

• no direct solution of (non-linear) CC equations possible => iterative schemes • rewrite CC equations

CCD: 0=( + )tab + Zab(tab) a b i j ij ij ij zeroth-order term higher-order terms • iterative sequence ab ab(n) ab(n+1) Zij (tij ) computed with tij = old amplitudes i + j a b new amplitudes

initial guess via MP2, convergence acceleration using DIIS CC equations: zeroth-order term

higher-order terms Computational Cost

target indices i,j,a,b mn ij tab = Zab || mn ij m

4 2 => total cost is 1/8 nocc * Nvirt CCD

4 2 2 4 3 3 linear terms: 1/8 nocc * Nvirt + 1/8 nocc * Nvirt + nocc * Nvirt => N6 computational cost

non-linear terms: mn ef tab tef = Zab N8 cost??? || mn ij ij m

linear terms CCD equations:

non-linear terms Factorization of Terms

mn ef tab tef = Zab || mn ij ij m N8 computational cost a) mn ef tef = Y mn || ij ij e 2 N6 computational cost Definition of Intermediates

• straightforward• ≈ implementation (term by term)

ab ef ab mn ij tmn + mn ef tij tmn || || 6 m

• via intermediates

intermediate = mn ij + mn ef tef Wmnij || || ij e

ab contraction mnij t W mn N6 cost m

1 CC energy: E = ij ab tab 4 || ij i,j a,b CC equations: eb ab 0= ab ij + P (ab) ae tij P (ij) mi tmj || F F e m 1 1 + tab + ab ef tef 2 Wmnij mn 2 || ij m,n e,f ae + P (ij)P (ab) mbej tim W intermediates: m e 1 1 ef ef = f + mn ef t mnij = mn ij + mn ef tij Fmi mi 2 || in W || 2 || n e,f e,f 1 af 1 bf = f mn ef t mbej = mb ej + mn ef t Fae ae 2 || mn W || 2 || jn m,n n f f Computational Cost

leading terms of CCD + CCSD

4 2 2 4 3 3 linear terms: 1/8 nocc * Nvirt + 1/8 nocc * Nvirt + nocc * Nvirt

4 2 3 3 non-linear terms: 1/8 nocc * Nvirt + nocc * Nvirt

4 2 2 4 3 3 total: 1/4 nocc * Nvirt + 1/8 nocc * Nvirt + 2 nocc * Nvirt

=> N6 computational cost less than 2 times as expensive as CID and CISD (linear terms only!) Standard CC Approaches

truncation of the cluster operator T:

cluster operator approximation cost

6 T=T1+T2 CCSD N 8 T=T1+T2+T3 CCSDT N 10 T=T1+T2+T3+T4 CCSDTQ N 12 T=T1+T2+T3+T4+T5 CCSDTQP N

… … …

T=T1+T2+ … +TN FCI Accuracy of CC Methods

deviation from FCI (in mH) for CO

CI CC SD 30.804 12.120 SDT 21.718 1.009 SDTQ 1.775 0.061 SDTQP 0.559 0.008 SDTQPH 0.035 0.002

calculations with cc-pVDZ basis, frozen core

E(FCI) = -113.055853 H

Historical Remarks

- CC theory Čížek, 1966

- first CC computations Čížek, 1966 (semiempirical, min. basis) Čížek, Paldus, Shavitt, 1972

Pople and co-workers, 1978 - CCD implementation Bartlett, Purvis, 1978

- CCSD implementation Purvis, Bartlett, 1982

- efficient closed-shell CCSD Schaefer and co-workers, 1987 Noga, Bartlett, 1987 - CCSDT implementation Scuseria and Schaefer, 1988 Oliphant, Adamowicz, 1991 - CCSDTQ implementation Kucharski, Bartlett, 1991 - general CC models Olsen, 2000, Kállay, Surján, 2001, Hirata, 2003 XI.

Approximate Treatment of Higher Excitations Need for Higher Excitations

High-Accuracy Calculations:

1 kJ/mol or better in thermochemical applications

0.1 pm accuracy in computed bond distances

... require considerations of excitations beyond singles and doubles

=> triples, quadruples, ... CCSDT Equations

H = EHF + fN + WN T = T1 + T2 + T3

1 E = E + 0 (f T ) + (W T ) + ((W T ) )T ) 0 HF | N 1 c N 2 c 2! N 1 c 1 c | projection on singly excited determinants additional term 0=S fN | due to T3

+(fN T1)c +(WN T1)c 1 1 +(f T ) +(W T ) + ((f T ) T ) + ((W T ) T ) N 2 c N 2 c 2 N 1 c 1 c 2 N 1 c 1 c 1 +((W T ) T ) + (((W T ) T ) )T ) + (W T ) 0 N 2 c 1 c 3! N 1 c 1 c 1 c N 3 c| CCSDT Equations

H = EHF + fN + WN T = T1 + T2 + T3

projection on doubly excited determinants

0= D WN | additional terms +(WN T1)c due to T3 1 +(f T ) +(W T ) + ((W T ) T ) N 2 c N 2 c 2 N 1 c 1 c 1 +((f T ) T ) + ((W T ) T ) + (((W T ) T ) T ) N 2 c 1 c N 2 c 1 c 3! N 1 c 1 c 1 c +(fN T3)c + (WN T3)c 1 1 1 + ((W T ) T ) + (((W T ) T ) T ) + ((((W T ) T ) T ) T ) 2 N 2 c 2 c 2 N 2 c 1 c 1 c 4! N 1 c 1 c 1 c 1 c +((W T ) T ) 0 N 3 c 1 c| CCSDT Equations

H = EHF + fN + WN T = T1 + T2 + T3 projection on triply excited determinants 0= (W T ) +(f T ) +(W T ) + ((W T ) T ) T | N 2 c N 3 c N 3 c N 2 c 1 c 1 1 +((f T ) T ) + ((f T ) T ) + ((W T ) T ) + ((W T ) T ) N 3 c 1 c 2 N 2 c 2 c 2 N 2 c 2 c N 3 c 1 c 1 1 + (((W T ) T ) T ) + ((W T ) T ) + (((W T ) T ) T ) 2 N 2 c 1 c 1 c N 3 c 2 c 2 N 2 c 1 c 1 c 1 1 + ((W T ) T ) T ) + ((((W )T ) T ) T ) T ) 0 2 N 3 c 1 c 1 c 3! N 2 c 1 c 1 c 1 c|

shorthand notation in the sense that all Ts are ((WN T2)cT1)c (WN T2T1)c connected to fN or WN CCSDT Equations

projection on triply excited determinants

0= T (WN T2)c +(fN T3)c +(WN T3)c | 1 +(W T T ) +(f T T ) + (f T 2) N 2 1 c N 3 1 c 2 N 2 c 1 1 + (W T 2) +(W T T ) + (W T T 2) 2 N 2 c N 3 1 c 2 N 2 1 c 1 1 +(W T T ) + (W T 2T ) + (W T T 2) N 3 2 c 2 N 2 1 c 2 N 3 1 c 1 + (W T T 3) 0 3! N 2 1 c |

„problems“ with CCSDT CCSD for comparison • computational cost : N8 • computational cost : N6 • storage of triples amplitudes: N6 • storage of doubles amplitudes: N4 two orders of magnitudes more expensive Iterative Approximations to CCSDT

additional approximations in the amplitude equations

- no storage of amplitudes for highest excitation

- reduced scaling in computational cost

=> CCSDT-n (n=1-3) Bartlett and co-workers (1984, 1987)

=> CC3 (within CCn hierarchy) Jørgensen and co-workers (1997) Iterative Approximations to CCSDT triples equations

0= (W T ) +(f T ) +(W T ) T | N 2 c N 3 c N 3 c 1 +(W T T ) +(f T T ) + (f T 2) N 2 1 c N 3 1 c 2 N 2 c 1 1 + (W T 2) +(W T T ) + (W T T 2) 2 N 2 c N 3 1 c 2 N 2 1 c 1 1 +(W T T ) + (W T 2T ) + (W T T 2) N 3 2 c 2 N 2 1 c 2 N 3 1 c 1 + (W T T 3) 0 3! N 2 1 c |

N8 terms: to be skipped in all cases Iterative Approximations to CCSDT reduced scaling

8 all N terms involve contractions with T3 in the triples equations

only remaining term with T3

abc (fNT3)c à T3 equation; ( + + + + ) t a b c i j k ijk scaling of remaining terms

terms in the T3 equations: N7 (dominates cost)

7 ((WN+ WNT1)T3)c à T2 equation: N (dominates cost)

6 (WN T3)c à T1 equation: N

overall cost : O(N7) per iteration Iterative Approximations to CCSDT

storage of T3 amplitudes

abc a(n) ab(n) (n+1) Z (t ,t ) tabc = ijk i ij ijk ( + + ) i j k a b c

compute the triples „on the fly“ independent of T3 followed by immediate calculation of T3 contribution to singles and doubles

(without storage) Iterative Approximations to CCSDT triples equations

0= (W T ) +(f T ) +(W T ) T | N 2 c N 3 c N 3 c 1 +(W T T ) +(f T T ) + (f T 2) N 2 1 c N 3 1 c 2 N 2 c 1 1 + (W T 2) +(W T T ) + (W T T 2) 2 N 2 c N 3 1 c 2 N 2 1 c 1 1 +(W T T ) + (W T 2T ) + (W T T 2) N 3 2 c 2 N 2 1 c 2 N 3 1 c 1 + (W T T 3) 0 3! N 2 1 c |

CCSDT-1: CCSDT-3: + + CCSDT-2: + CC3: + Accuracy of Approximate Treatments of Triples

deviation from FCI and CCSDT (in mH) for CO

Δ(FCI) Δ(CCSDT) CCSD 12.12 11.11 CCSDT-1 0.13 -0.88 CCSDT-2 1.52 0.51 CCSDT-3 1.47 0.46 CC3 0.12 -0.89

calculations with cc-pVDZ basis, frozen core Non-Iterative Approximations to CCSDT two-step procedure:

• perform CCSD calculation

T = T + T 1 2 • add perturbative corrections due to T 3

E = E(CCSD) + ΔE(T)

use of converged T1 and T2 amplitudes

→ CCSD+T(CCSD), CCSD(T), .... Coupled-Cluster and Perturbation Theory

perturbative expansion of CC energy and equations

(0) (1) (2) H0 = EHF + fN E = E + E + E + ... (1) (1) (2) (3) H = WN T = T + T + T + ...

- recovers Møller-Plesset (MP) perturbation theory - CCSD contains all second- and third-order terms, lacks some higher-order terms

- CC sums certain terms (via iterative solution) to infinite order - correct CC energy for missing terms using perturbation theory

Coupled-Cluster and Perturbation Theory perturbative triples correction to energy

(2) (2) E = 0 T † f T 0 T | 3 N 3 |

second-order T3, from MP4

compute T3 with converged CCSD amplitudes

E = 0 T † f T 0 T | 3 N 3 | defines CCSD+T(CCSD)

Urban, Noga, Cole, Bartlett, 1985 Coupled-Cluster and Perturbation Theory

additional consideration of a singles contribution

(2) (2) E = 0 T † W T 0 | 1 N 3 | fifth-order term! triples correction to energy (converged amplitudes)

E = 0 (T † f + T † W ) T 0 T | 3 N 1 N 3 | defines CCSD(T)

Raghavachari, Trucks, Pople, Head-Gordon, 1989 CCSD(T)

perturbative corrections on top of CCSD

• fourth-order contribution abc 2 1 wijk ET (4) = | | 36 (i + j + k a b c) i,j,k a,b,c

wabc = P (ij/k)P (ab/c) bc ek tae mc jk tab ijk || ij || im e m computed with CCSD amplitudes • fifth-order contribution abc abc 1 wijk wijk ET (5) = 36 (i + j + k a b c) i,j,k a,b,c wabc = P (ij/k)P (ab/c) bc jk ta ijk || i permutation operator: P (pq/r) Z(pqr)=Z(pqr)+Z(qrp)+Z(rpq) CCSD(T) computational cost

wabc = P (ij/k)P (ab/c) bc ek tae mc jk tab ijk || ij || im e m one summaton index (e or m) 3 4 => cost scale with nocc Nvirt six target index (i,j,k,a,b,c)

CCSD(T): cost N6 per iteration, N7 for non-iterative step

no need to store T3 amplitudes

„gold standard“ of quantum chemistry Accuracy of Approximate Treatments of Triples

deviation from FCI and CCSDT (in mH) for CO

Δ(FCI) Δ(CCSDT) CCSD 12.12 11.11 CCSDT-1 0.13 -0.88 CCSDT-2 1.52 0.51 CCSDT-3 1.47 0.46 CC3 0.12 -0.89 CCSD(T) 1.47 0.46 CCSD+T -0.05 -1.06

calculations with cc-pVDZ basis, frozen core Further Developments perturbative triples • Λ-CCSD(T) Crawford, Stanton, 1999 Kucharski, Bartlett, 1999 replacesreplaces T3† byby 3 slightly more stable • CCSD(T-n), n=2,3,4,... Eriksen et al., 2014 most rigorous treatment n=4 required to match CCSD(T) perturbative quadruples • CCSDT(Q) Bomble et al., 2005 non-iterative N9 analogue to CCSD(T)

• CCSDT(Q-n), n=2,3,4,... Eriksen et al., 2015 most rigorous treatment XII.

Computational Thermochemistry: A Perfect Playground for CC Theory Computational Thermochemistry

quantum-chemical calculation of thermochemical energies via energy differences most often computed:

atomization energies heats of formation calculation involves: electronic part non-electronic part

e.g. vibrational corrections Chemical Accuracy accuracy required for realistic chemical predictions

1 kcal/mol ˆ= 4.184 kJ/mol 4.33 . 10-2 eV

350 cm-1 1.59 mHartree

sub-chemical accuracy: 1 kJ/mol and better

density-functional theory not accurate enough Approximate Quantum Chemistry

FCI

FCI/infinite basis CCSDTQ CCSDT CC CCSD(T) correlation

CCSD

HF-SCF electron

cc-pVDZ cc-pVTZ cc-pVQZ cc-pV5Z ….

basis set Approximate Quantum Chemistry

FCI

FCI/infinite basis CCSDTQ CCSDT CC CCSD(T) correlation

CCSD

HF-SCF electron

cc-pVDZ cc-pVTZ cc-pVQZ cc-pV5Z ….

basis set Basis-Set Extrapolation: HF Energy

E

-112.780 extrapolation of HF-SCF energy

-112.785

-112.790 TZ QZ 5Z 6Z

E(HF-SCF/aug-cc-pVXZ) = E1(HF-SCF) + a exp( bX)

a, b, and E∞(HF-SCF) from extrapolated value 3 HF-SCF energies

D. Feller, J. Chem. Phys. 98, 7059 (1993) Basis-Set Extrapolation: Correlation Energy

E

-0.405 extrapolation of CC energy

-0.410

-0.415

QZ 5Z 6Z 7Z 8Z c E1(CC) = E(CC/aug-cc-pVXZ) X3

c and ΔE∞(CC) from 2 CC energies extrapolated value

Helgaker et al., J. Chem. Phys. 106, 9639 (1997) High-accuracy Extrapolated Ab-initio Thermochemistry

HF-SCF extrapolated (Q56) + ΔCCSD(T) extrapolated (ae, 56) + ΔCCSDT extrapolated (fc, TQ) + ΔCCSDTQ fc, cc-pVDZ + ΔCCSDTQP fc, cc-pVDZ + scalar relativistic CCSD(T)/aug-cc-pCVTZ + SO splittings MRCISD/cc-pVDZ+PP + ZPE (anharm.) CCSD(T)/cc-pVQZ + DBOC CCSD/aug-cc-pCVQZ

Tajti et al., J. Chem. Phys. 121, 11599 (2004)

Bomble et al., J. Chem. Phys. 125, 064108 (2006) Harding et al., J. Chem. Phys. 128, 114111 (2008) HEAT: Atomization Energy of CO

HF-SCF 730.13 extrapolated (Q56) Δ CCSD(T) 356.53 extrapolated (ae, 56)

ΔCCSDT -2.37 extrapolated (fc, TQ) ΔCCSDTQ 2.22 fc, cc-pVDZ ΔCCSDTQP 0.13 fc, cc-pVDZ scalar relativistic -0.65 CCSD(T)/aug-cc-pCVTZ SO splittings -1.20 MRCISD/cc-pVDZ+PP ZPE (anharm.) -12.99 CCSD(T)/cc-pVQZ DBOC 0.02 CCSD/aug-cc-pCVQZ

Sum (Theory) 1071.83 all energies in kJ/mol Experiment 1072.13±0.09 Harding et al., J. Chem. Phys. 128, 114111 (2008) Statistical Evaluation

High-accuracy Extrapolated Ab-initio Thermochemistry atomization energies of

N2, H2, F2, CO, O2, C2H2, CCH, CH2, CH, CH3,

CO2, H2O2, H2O, HCO, HF, HO2, NO, OH

mean error max. error RMS error -0.24 0.87 0.42

Harding et al., J. Chem. Phys. 128, 114111 (2008) XIII.

Closed- and Open-Shell CC Methods Closed- and Open-Shell Systems

↑ ↑ ↑↓ ↑ ↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ ↑↓ high-spin low-spin closed shell open-shell open-shell multi- ⇒ RHF reference ⇒ UHF, ROHF reference ⇒ determinantal description

spin-adapted spin-orbital no routine CC formalism CC formalism CC treatment Closed-Shell CC Approaches restricted HF reference

1 ¯ ¯ 0 ˆ= 11 ... N/2N/2 | N! | | doubly occupied spatial orbitals α spin orbital β spin orbital

=> relationships among two-electron integrals pq rs = p¯q¯ r¯s¯ = pq¯ rs¯ = ... | | | pq rs = pq¯ rs¯ pq¯ sr¯ || | | Closed-Shell CC Approaches

non-vanishing spin cases for amplitudes

¯ ¯ ¯ ¯ ¯ ta = ta¯ tab = ta¯b tab = tba = tba = tab i ¯i ij ¯i¯j i¯j ¯ji i¯j ¯ji

αα ββ αααα ββββ αβαβ βαβα αββα βααβ

ab a¯b ba¯ a¯b ba¯ in addition t = t ¯ t ¯ t ¯ = t ¯ ij ij ij ij ji

only αα (singles) and αβαβ (doubles) required Closed-Shell CC: Spin Integration closed-shell CCD • consider only αβαβ amplitude equations ¯ 0= ab exp( T ) H exp(T ) 0 i¯j | 2 2 | • consider only non-zero contributions based on spin cases ¯ ¯ + teb + te¯b + ... ··· Fae i¯j Fae¯ i¯j e e • rewrite all terms using αα and αβαβ quantities ae ae¯ ¯ ¯ t + ¯ ¯ t Wmbej im Wm¯ be¯j im¯ 1 ae¯ ea¯ 1 ea¯ = (2 ¯ ¯ + ¯ ) (2 t t )+ ¯ t 2 Wmbej Wmbej¯ im¯ im¯ 2 Wmbej¯ im¯ => CCD equations in terms of f , p¯q r¯s , and tab pq | ij Closed-Shell CCD

• amplitude equations e¯b a¯b 0= a¯b i¯j + P (ab) aet ¯ P (ij) mi t ¯ | F ij F mj e m a¯b ef¯ + ¯ t + a¯b ef¯ t Wmni¯ j mn¯ | i¯j m,n e,f 1 ae¯ ae¯ + P (ia, jb) (2 ¯ ¯ + ¯ ) (2 t t ) 2 + Wmbej Wmbej¯ im¯ m¯i m e 1 ae¯ ae¯ + ¯t ¯ t 2 Wmbej¯ m¯i Wmbei¯ m¯j • actual (optimal) computational cost 1 1 n4 N2 + 4n3 N3 + n2 N4 2 occ virt occ virt 4 occ virt • automatically spin adapted Open-Shell CC Approaches: UHF-CC unrestricted HF reference different spatial 1 0 ˆ= ... ¯ ... ¯ parts for α and 1 N 1 N | (N + N)! | | β spin orbitals non-vanishing spin cases for amplitudes

¯ ¯ ¯ ¯ ¯ ta ta¯ tab ta¯b tab = tba = tba = tab i ¯i ij ¯i¯j i¯j ¯ji i¯j ¯ji

αα ββ αααα ββββ αβαβ βαβα αββα βααβ

αα and ββ (singles) and αααα, ββββ, and αβαβ (doubles) required Open-Shell CC Approaches: UHF-CC spin-integrated UHF-CCD equations

αααα /ββββ spin cases

eb ab 0= ab ij + P (ab) aetij P (ij) mi tmj | F F e m + tab + ab ef tef Wmnij mn | ij m

αβαβ spin case

e¯b ae¯ a¯b a¯b 0= a¯b i¯j + t + ¯ t t ¯ t | Fae i¯j Fbe¯ i¯j Fmi m¯j Fm¯ j im¯ e e¯ m m¯ a¯b ef¯ + ¯ t + a¯b ef¯ t Wmni¯ j mn¯ | i¯j m,n e,f ae ae¯ e¯b + ¯ ¯t + ¯ ¯t + t Wmbej im Wm¯ be¯j im¯ Wmaei m¯j m e m¯ e¯ m e e¯¯b e¯b ae¯ + t + ¯t + ¯ t Wma¯ ei¯ m¯ ¯j Wmae¯ j im¯ Wmbei¯ m¯j m¯ e¯ m¯ e m e¯ actual (optimal) computational cost 5 3 n4 N2 + 20 n3 N3 + n2 N4 2 occ virt occ virt 2 occ virt 4-5 times more expensive than RHF-CCD/CCSD Open-Shell CC Approaches: ROHF-CC

restricted open-shell HF (ROHF) reference same spatial 1 ¯ ¯ parts for α and 0 ˆ= 1 ... N 1 ... N | (N + N)! | | β spin orbitals αα and ββ (singles) and αααα, ββββ, and αβαβ (doubles) required actual (optimal) computational cost

4 2 3 3 2 4 2.5 nocc Nvirt + 20 nocc Nvirt + 1.5 nocc Nvirt

essentially the same cost as UHF-CC Spin-Orbital Based CC Methods

• T contains spin-orbital excitations • no spin properties enforced

• since S2,T =0in case of open-shell systems ⇒ truncated CC wavefunctions no spin eigenfunctions

• UHF-CC

2 2 S UHF = s(s + 1) and S = s(s + 1) CC • ROHF-CC

2 2 S ROHF = s(s + 1) but S = s(s + 1) CC Spin-Adaption in CC Theory

• generators of the unitary group

2 a a S ,Ei =0 with Ei = aa† ai =, • closed-shell CC summation - T can be rewritten in terms of a Ei over spins - spin adaptation trival • open-shell CC - a usage of E i leads to non-commuting excitation operators T = ta Ea,... a b a b 1 i i => Ei ,Ej = ibEj ajEi i a singly occupied orbital appears as occupied and virtual Current Status of Open-Shell CC Theory

• spin-orbital approaches (UHF, ROHF reference) - standard choice • unitary-group based approaches Janssen, Schaefer, 1991 - rigorously spin adapted Li, Paldus, 1994 - complicated, automatized implementations • partially spin-adapted approaches Janssen, Schaefer, 1991 - open-shell orbital(s) not spin adapted Knowles, Hampel, Werner, 1993 - much simpler Neogrády, Urban, Hubač, 1994 • spin-restricted approaches - exact CC spin-expectation value Szalay, Gauss, 1997 • spin-free combinatoric approaches - unitary group based, spin adapted Datta, Mukherjee, 2008 - modified exponential ansatz XIII.

Analytic Energy Derivatives in CC Theory CC Lagrangian

• CC theory is nonvariational

- direct differentiation of CC energy involves derivatives of amplitudes

=> computationally inefficient!!

- efficient formulation of CC gradients via so-called CC Lagrangian

• CC Lagrangian Lagrange multipliers

L = 0 exp( T )H exp(T ) 0 + exp( T )H exp(T ) 0 CC | | I I | | I energy CC equations as constraints Lambda Operator

L = 0 exp( T )H exp(T ) 0 + exp( T )H exp(T ) 0 CC | | I I | | I compact notation L = 0 (1 + ) exp( T ) H exp(T ) 0 CC | |

Λ deexcitation operator = 1 + 2 + ...

i 1 ij = a†a = a†a a†a 1 a i a 2 4 ab i a j b i a i,j a,b

diagrammatic representation Stationarity Conditions wrt λ-amplitudes

LCC =0 0= I exp( T )H exp(T ) 0 I | | CC equations wrt t-amplitudes

LCC =0 0= 0 (1 + ) (exp( T )H exp(T ) E) I tI | | Λ equations stationarity => (2n+1) rule for t-amplitudes and (2n+2) for λ-ampliutudes CC Gradients

• differentiate CC Lagrangian wrt perturbation x

L = 0 (1 + ) exp( T ) H exp(T ) 0 CC | |

• exploit (2n+1) and (2n+2) rule

E H = 0 (1 + ) exp( T ) exp(T ) 0 x | x |

no perturbed t- and λ-amplitudes required! CCD Gradients

• cluster operator T = T2

• Lambda operator = 2

• CCD Lagrangian

L = 0 (1 + ) exp( T ) H exp(T ) 0 CCD | 2 2 2 | • CCD amplitude equations 0= exp( T ) H exp(T ) 0 D| 2 2 | • CCD lambda equations 0= 0 (1 + ) (exp( T ) H exp(T ) E ) | 2 2 2 CCD | D Λ Equations for CCD: Diagrams Λ Equations for CCD: Algebraic Form

(perturbation independent) linear equations for λ amplitudes Density Matrices in CC Gradient Theory introduction of density matrices

E H = 0 (1 + ) exp( T ) exp(T ) 0 x | x | f 1 pq rs = D pq + || pq x 4 pqrs x p,q p,q,r,s

reduced two-particle reduced one-particle density matrix

density matrix 0 (1 + ) exp( T ) a† a†a a exp(T ) 0 | { p q s r} | 0 (1 + ) exp( T ) a† a exp(T ) 0 | { p q} | gradients in terms of integral derivatives and density matrices Diagrams for CCD Density Matrices

• one-particle density matrix

• two-particle density matrix Algebraic Expressions for CCD Density Matrices

• one-particle density matrix 1 1 D = jm tef D = mn tbe ij 2 ef im ab 2 ae mn m m,n e e,f • two-particle density matrix 1 1 = kl tef = mn tcd ijkl 2 ef ij abcd 2 ab mn m,n e,f im be ij ajib = ae tjm abij = ab m e ab 1 mn ef ab 1 mn ae bf ijab = tij + ef tij tmn + P (ij)P (ab) ef tim tjm 4 2 m,n m,n e,f e,f 1 mn eb af 1 mn ef ab P (ab) ef tmn tij P (ij) ef tmj tin 2 2 m,n m,n e,f e,f CC Gradients with Orbital Relaxation extended Lagrangian ≡ CC Lagrangian + orbital constraints

L = 0 (1 + ) exp( T ) H exp(T ) 0 CC | | Lagrange CC energy and CC equations multipliers

+2 D f + I c S c ai ai pq p µ q pq a i p,q µ, Brillouin condition orthonormality of MOs

stationarity conditions ⇒ elimination of MO derivatives CC Gradients with Orbital Relaxation parameterization of orbital changes

cµp = cµq Tqp matrix T q stationarity requirements general (non-unitary) transformation

LCC (1) =0 = Iab Tab LCC (2) =0 = Iai + Iia Tia

LCC (3) =0 = Z-vector equations for Dai Tai

LCC (4) =0 = Iij Tij Z-Vector Equations for CC Gradients

Z-vector equations

D [ ae im + am ie + ( )] = X em || || ae im a i ai m e

with 1 X = ( ap qr + qr ap ) ai 4 ipqr || qrip || p,q,r 1 ( ip qr + qr ip ) 4 apqr || qrap || p,q,r 1 + D ( pa qi + pi qa ) 2 pq || || p,q

linear equations for orbital relaxation contribution to Dai CC Gradients with Orbital Relaxation differentiation of „extended“ CC Lagrangian

Hamiltonian with AO derivative integrals L CC = 0 (1 + ) exp( T ) Hx exp(T ) 0 x | |

h µ +2 D c µ + DSCF || c ai µa x x i i a µ, ,

Sµ + I c c pq µp x q p,q µ, AO derivative integrals contains no derivative of wavefunction parameters AO Formulation of CC Gradients differentiation of „extended“ energy functional

E 1 = D f (x) + pq rs x + I Sx x pq pq 4 pqrs || pq pq p,q p,q,r,s p,q integral derivatives contain no MO derivative contributions reformulation in AO representation

E hµ SCF µ 1 µ Sµ = D + D || + || + Iµ x µ x x 4 µ x x µ , µ,,, µ,

density matrices and intermediates back transformed to AO basis Implementation of CC Gradients required steps for gradients

CC and Λ equations (∼ N6,N8,…)

CC density matrices (∼ N6,N8,…)

MO → AO transformation of density matrices (∼ N5)

contraction with integral derivatives (no storage, ∼ N4)

computational cost do not scale with Npert Historical Remarks

- CC Lambda operator Arponen, 1983 Adamowicz, Bartlett, 1984 - Z-vector method Handy, Schaefer, 1984

- Lagrangian technique Jørgensen, Helgaker, 1988

- CCSD gradients Schaefer and co-workers, 1987

- open-shell CCSD gradients Gauss, Stanton, Bartlett 1991

- CCSD(T) gradients Scuseria, 1991, Lee, Rendell 1991 Watts, Gauss, Bartlett, 1992

- CCSDT gradients Gauss, Stanton 2001

- general CC gradients Kállay, Szalay, Gauss, 2003 Accuracy of CC Geometrical Parameters

calculated r(OH) for H2O (in Å) CI CC SD 0.96131 0.96435 SD(T) 0.96575 SDT 0.96251 0.96583 SDTQ 0.96593 0.96614 SDTQP 0.96606 0.96616 SDTQPH 0.96616 0.96616 FCI 0.96616

calculations with cc-pVDZ basis XIV.

Availability and Applicability Standard CC Approaches

N6 CCSD many programs such as

ACES2, ACES3, CFOUR, DALTON, GAUSSIAN, GAMESS, MOLCAS, MOLPRO, NWCHEM, CCSD(T) 7 N ORCA, PSI, PySCF, Q-CHEM, TURBOMOLE, ...

CCSDT N8 ACES2, CFOUR, NWCHEM

CCSDT(Q) N9 CFOUR, MRCC CCSDTQ N10

... XV.

What is Still Missing? Further Topics (not covered)

large molecules (linear scaling) multi-reference treatments e.g., local CC methods (à F. Neese) e.g., Mk-MRCC, ic-MRCC (not routine!) explicitly-correlated CC methods properties via CC methods R12-CC and F12-CC analytic CC derivatives response theory (à T. Helgaker, J. Olsen) higher excitations in CC theory general CC excited states via CC methods automatized implementation equation-of-motion (EOM) CC string-based methods CC response theory (à Friday)

other open-shell CC methods relativistic effects EOM-IP-CC 4- and 2-component CC (à L. Visscher) Fock-space CC