Equation-of-Motion Coupled-Cluster Methods for Open-Shell and Electronically Excited Species

Anna I. KrylovUniversity of Southern California, Los Angeles

IMA WorkshopMinneapolis, 2008

Recent review: Krylov, Equation-of-Motion Coupled-Cluster Methods for Open-Shell and Electronically Excited Species: The Hitchhiker’s

Guide to Fock Space, Ann. Rev. Phys. Chem. 59, 433 (2008).

Outline:

1.CI and truncated CI: overview.2.Size-extensive alternative: Coupled-cluster ansatz.3. Extension to electronically excited and open-shell states: EOM-CC formalism.4. EOM-CC advantages and limitations. 5. Examples.6. More rigorous derivation of EOM-CC equation.

Exact solution of the Schroedinger equation and configuration interaction (CI) approach1.Chose one-electron basis set (M orbitals), construct reference determinant (N orbitals) 0

2. Linear ansatz for the wave-function: =(1+C1+C2+ … CN)0, where C1=cia a+i

Thus, operators C1..CN will generate all possible distributions of N electrons over M orbitals

3. Energy functional: E=<|H|>/<|>Amplitude equations: Variational Principle -> CI eigen-problem for the ground and excited states: HC=CE

Exact solution (FCI). Approximations: truncated CIHierarchy of approximations: HF->CISD->CISDT->….->FCIProblems with truncated CI: violation of size extensivity

For non-interacting systems (HAB=HA+HB): EAB=EA+EB and AB=AxB

But: ACISDxB

CISD is not equal ABCISD

Solution: Coupled-cluster ansatz: =eT0=(1+T + ½T2 + 1/6 T3 + … ) 0, T1=ti

a a+i

If T=T1+T2 : - higher-order excitations are present, e.g., (T1)2, (T2)2,(T2)2T1, etc- size-extensivity is satisfied: eTAxeTB=eTA+TB (provided that T is strictly an excitation operator wrt the reference vacuum);- but exponential expansion does not terminate in energy functional E=<|H|>/<|> for variational principle.

Equations (projection principle): E=<0|e-T H eT | 0> <|e-T H eT -E| 0>=0Can be recast in a more general variational form and extended to excited states.

Hierarchy of approximations to the exact wave function: Single-reference models for the ground stateSCF : nex=R1(CIS)MP2 : SCF + T2 by PT CIS + R2 by PT [CIS(D)]CCSD: exp(T1+T2) ex=(R1+R2)(EOM-CCSD)CCSD(T): CCSD + T3 by PTCCSDT: exp(T1+T2+ T3) ex=(R1+R2+R3)(EOM-CCSDT)…………………………………………………................................. FCI: =(1+T1+T2 + … +Tn )- exact!

T1=ia tia a+ j T2=0.25*ijab tij

ab a+b+ ji

Why open shells and electronically excited statesare difficult?

Electronic degeneracy -> multi-configurational wavefunctions

The CC hierarchy of approximations breaks down.

Equation-of-motion theory and specific models:Rowe, Rev. Mod. Phys. 40, 153 (1968) Simons, Smith, JCP 58, 4899 (1973) McWeeny, “Methods of Molecular Quantum Mechanics”Lowdin, “Some aspects of the Hamiltonian and Liouvillian Formalism …”,AQC 17, 285 (1985)Sekino, Bartlett, IJQCS 18 255 (1984)Stanton, Bartlett, JCP 98, 7029 (1993)Stanton, Gauss, JCP 101, 8938 (1994)Nooijen, Bartlett, JCP 102, 3629 (1995)Wladyslawski, Nooijen, ACS series 828, 65 (2002)

Also works of Mukherjee, Pal, Piecuch, Emrich, McKoy, Kutzelnigg, Kaldor, Werner, Hirata.Related: linear response works of Koch, Jorgen, Jensen, Jorgensen, Head-Gordon, Lee, Korona, etc.SAC-CI by Nakatsuji.Recent developments: Crawford, Piecuch, Kowalsky. AND MUCH MORE!!!Recent review: Krylov, Equation-of-Motion Coupled-Cluster Methods for Open-Shell and Electronically Excited Species: The Hitchhiker’s Guide to Fock Space, Ann. Rev. Phys. Chem. 59, 433 (2008).

Equation-of-Motion Coupled-Cluster Methods

1. regardless of T, has same spectrum as H

2.=0 + R10 + R20 + ......, where Rn is some general excitation operator, e.g., R1=ri

a a+i (or R1=ra a+ or R1=rii, etc)

3. Apply bi-Variational Principle: EOM eigen-problem

4. Specific EOM model: choice of excitation operators T, R, and the reference 0

)exp()exp( THTH

00 ERRH

E0 X X

0 X X

0 X X

0 1 2

0

1 2

0|| 0 Hex

EOM models: Choice of T

1. Excitation level, e.g., T=T1+T2

2. Amplitude equations. If T satisfies CC equations,

H

- size extensivity- compact wf-s, e.g., exact=0

- correlation effects are "wrapped in": same scaling but higher accuracy

EOM-IP: (N) =R(-1)0(N+1)

i ija

EOM-EE: (Ms=0) =R(Ms=0)0(Ms=0)

ia ij

ab

EOM-EA: (N) =R(+1)0(N-1)

a iab

ia

EOM-SF: (Ms=0)=R(Ms=-1)0(Ms=1)

EOM MODELS: CHOICE OF R and 0

Size-extensivity: Be exampleBe /6-31G*

State FCI CISD SF-CISD EOM-CCSD1S(1s22s2) 0.0 0.0 0.0 0.03P(1s22s2p) 2.86 2.88 2.86 2.861P(1s22s2p) 6.58 6.60 6.58 6.58

State CISD SF-CISD EOM-CCSD

Be(1S)Ne(1S) 0.0 0.0 0.0

Be(3P)Ne(1S) 6.67 2.86 2.86

Be(1P)Ne(1S) 9.39 6.58 6.58

Be @ USC & Ne @ Berkeley

Krylov, CPL 350, 522 (2001); Sears, Sherrill, Krylov, JCP, 118, 9084 (2003)

Equation-of-Motion Coupled-Cluster Theory

Truncated EOM model, e.g., EOM-CCSD: diagonalize H-bar in the basis of singly and doubly excited determinants (amplitudes T - from CCSD equations).

Same cost as CISD (N6) but: - size-intensive - has higher accuracy (correlation is “wrapped in” through the similarity transformation). Multistate method, describes degeneracies and near-degeneracies, as well as interacting states of different nature (just because it is diagonalization problem).Bi-variational formulation facilitates properties calculations(Hellmann-Feynman theorem).

Limitations of CCSD and EOM-CCSD:

1.Scaling is N6.

2. Need triples corrections for “chemical accuracy”, e.g.CCSD(T) (scales N7).3. Not always possible to find a well-behaved reference fromwhich target states of interest can be accessed via singleexcitations -> cannot describe global potential energy surfaces. 4. Non-hermitian nature sometimes causes problems (e.g.,wrong dimensionality of conical intersections).

Excited states of sym-triazine

*n manifold * manifold

Rsn manifold Rpn manifold

Mozhayskiy, Babikov, and Krylov, JCP 124, 224309 (2006)Schuurman and Yarkony, JCP 126, 044104 (2007)

- All low-lying excited states of Tz involve transitions from/to degenerate MOs. - Some are Jahn-Teller distorted and some exhibit glancing-like intersections.- EOM-EE describes accurately these manifolds, interactions, and degeneracies.

Excited singlet states of sym-triazine

Neutral geometry

Cation geometry

EOM-CCSD/6-311++G**

EOM for the [Cs..Tz]+ complex

Initial CT state

Single excitations of initial CT state

Double excitations

of initial CT state

Single excitations of the reference state

Initial charge transfer state

Final charge transfer states

EOM reference: [Cs+..Tz]

Cs · · Tz+ Cs+ · · Tz*

Equation-of-motion formalism

Consider the Hamiltonian H (non-hermitian), and its exact eigenstates |0> & |f>:H|0>=E0|0> & H|f>=Ef|f>

Consider excitation operator R(f): R(f)|0> =|f> Define R(f) as: R(f)=|f><0|For any |0> (<0|0> non-zero): R(f)|0>=|f><0|0> => [H,R(f)]|0> = of R(f)|0> (*) - no knowledge of exact initial and final states is necessary to determine exact of

- (*) is an operator equation, valid not only in a subspace. Define de-excitation operator L(f): L(f)=|0><f|

EOM functionals:of=<0|L(f)[H,R(f)]|0> / <0|L(f)R(f)|0>

of=<0|[L(f),[H,R(f)]]-|0> / <0|[L(f),R(f)]-|0> of=<0|[L(f),[H,R(f)]]+|0> / <0|[L(f),R(f)]+|0> ALL three functionals: - exact result when R, L is expanded over the complete basis set

- identical equations when the killer condition is satisfied: L(f) |0> = 0 (reference is “true” vacuum)

or L(f) |0> = |0><f|0> = 0 (reference is orthogonal to final states)

EOM equations

1. Expand R, L over a finite operator basis set:

R(f) = k rk k

L(f)=k lk k

2. Use bivariational priciple with any of the three funcionals, arrive to the matrix equations:

(H-E0)R=R L(H-E0)=L E0=<0|H|0>