Approximations to the exact solution of the Schroedinger ...

Approximations to the exact solution of the Schroedinger equation (in the finite one-electron basis set).

Ψ=(1+C1+C2+ … CN)Φ0, where C1Φ0=ΣciaΦi

a

1.Crudest approximation: use only one determinant Φ0 = det{φ1,…, φn}Variational principles tells us how to choose “best”set oforbitals to make the best Φ0: Hartree-Fock modelNon-linear equations for orbitals: F(φ)φi(x)=εiφi(x)2. Beyond HF: Truncated CI CISD: Ψ=(1+C1+C2)Φ0 include Φ0 and all singly and doubly excited determinants

Hierarchy of approximations: HF->CISD->CISDT->….->FCINot size-extensive!

Tuesday, November 25, 2014

Truncated CI:

Good: 1.Variational: we know that we have an upper bound on the ground-state energy (there is a much weaker result for excited states).

2. Systematic convergence: include higher excitations -> larger expansion -> closer to the exact answer. We knowformally that our hierarchy of approximations HF->CISD->CISDT->….->FCI works.

3. Linear parameterization: simple diagonalization.

4. “Black box”: no system-dependent parameterization.


Truncated CI:

Not so good:

Size of the problem scales rather steeply: CISD: N6

CISDT: N8

CISDTQ: N10

For example, can do GFP chromophore (114 el) with CISD,but forget about CISDT or doubling size of the system.


Truncated CI:

Really bad: Violation of size-extensivity (and size-intensivity)

For non-interacting systems A and B:HAB=HA+HB EAB=EA+EB and ΨAB=ΨAxΨB

But: ΨACISDxΨB

CISD is not equal ΨABCISD

One H2 (hydrogen) molecule: exact energy Two H2 molecules: not exactFive H2: junk.


EHF nxEHF(H2) ECISD nxECISD Error

H2 -1.132076 -1.171015

2H2 -2.264151 -2.264152 -2.340775 -2.342030 0.001255

3H2 -3.396227 -3.396228 -3.509410 -3.513045 0.003635

5H2 -5.660378 -5.660380 -5.843722 -5.855075 0.011353(7 kcal/mol)!

Non-interacting H2 molecules (separated by 100 A)total Hartee-Fock and CISD ground-state energies(in atomic units)

cc-pVTZ basis, RHH=0.7 A

Compare: interaction energy of two H2 molecules is0.1 kcal/mol


Consequences of violation of size-extensivity

1. Errors in interaction energies: E(AB) versus E(A) + E(B)

2. Errors due to increasing size of molecules, e.g., cannotcompare homological series: CH3(CH2)nCH3

3. Accuracy for small molecules deteriorates in largermolecules -> benchmarks are not transferable


3. Coupled-cluster ansatz Ψ=eTΦ0=(1+T + ½T2 + 1/6 T3 + … ) Φ0,

where T1=Σtia a+i or T1Φ0=Σti

aΦia

If T=T1+T2 (CCSD, analogue of CISD) 1. Higher-order excitations are present, e.g., (T1)2, (T2)2,(T2)2T1, etc

2. Size-extensivity is satisfied: eTAxeTB=eTA+TB provided that T is strictly an excitation operator wrt the reference vacuum Φ0, in this case TA and TB commute: [TA,TB]=0.


Coupled-cluster equations: how do we find amplitudes T? Ψ=eTΦ0=(1+T + ½T2 + 1/6 T3 + … ) Φ0

Exponential expansion does not terminate (soon enough) in energy functional E=<Ψ|H|Ψ>/<Ψ|Ψ> for variational principle to be feasible (there is variational CC code developed fortoy purposes, it is more expensive than FCI).

Equations (projection principle): E=<Φ0|e-T H eT | Φ0> <Φµ|e-T H eT -E| Φ0>=0

Can be recast in a more general variational form and extended to excited states.


Example: CCD equations

GFP example:i,j: 0,….115a,b: 0,….3002.94x108 variables

Solve iteratively:X=DA

-1(A’X + BX2)

Use DIIS

Converges 10-20 iter-swith convergence 10-8

(norm change below threshold)


Coupled-cluster methods:1. Rigorously size extensive.2. CCSD is exact for 2 electrons, CCSDT – exact for 3, etc.3. Non-variational: does not provide an upper bound of the exact energy (but satisfy some other VP).4. Equations are similar to CI but have additional non-linear(in amplitudes T) terms. Solved iteratively for one solutiononly (there are studies of other solutions of CCSD equations. Unlike CI, these solutions do not provide meaningful descriptionof the excited states).5. Computational scaling is the same as for the analogous CI (i.e., CCSD scales as N6, CCSDT – as N8, etc) 6. Well defined model (satisfy Pople’s criteria), systematicallyimprovable.7. Numeric performance: superior to CI


Hierarchy of approximations to the exact wave function: Single-reference models for the ground stateSCF : Ψ=Φ0=|ϕ1...ϕn> Ψex=R1 Ψ0 (CIS) MP2 : SCF + T2 by PT CIS + R2 by PT [CIS(D)]CCSD: Ψ=exp(T1+T2) Φ0 Ψex=(R1+R2)Ψ0 (EOM-CCSD) CCSD(T): CCSD + T3 by PTCCSDT: Ψ=exp(T1+T2+ T3) Φ0 Ψex=(R1+R2+R3)Ψ0 (EOM-CCSDT) …………………………………………………................................. FCI: Ψ=(1+T1+T2 + … +Tn )Φ0 - exact!

Φ0 Τ1Φ0Τ2Φ0

T1=Σia tia a+ j T2=0.25*Σijab tijab a+b+ ji

N

N3-N4, iterativeN5, non-iterativeN6, iterativeN7, non-iterativeN8, iteratrive


Distribution of errors for calculated bond lengths in closed-shell species

Helgaker et al., JCP 106 6430 (1997)

Basis

Cor

rela

tion


13

Summary:1. Combination of PT and CC theory: hierarchy of size-extensive approaches to electron correlation.2. MP2: 80% of correlation energy, CCSD: 95%, CCSD(T) - 99%. 3. Predictive, black-box approaches.4. Importance of basis set.5. Scope of applicability: when HF is qualitatively correct and describes leading character of the wave function.


Formulation of coupled-cluster theory using similarity transformed Hamiltonian:

- variational properties of CC (important for gradients and properties) - extension to the excited states (remember, HΨL=ELΨL)

Why excited states are difficult? Multi-configurational character; problems with truncated CI.


Similarity transformation and coupled-cluster methods

1. regardless of T, has the same spectrum as H

2. Ψ=Φ0 + R1Φ0 + R2Φ0 + ......, where Rn is some general excitation operator, e.g., R1=Σri

a a+i (or R1=Σra a+ or R1=Σrii, etc)3. Apply bi-Variational Principle: EOM eigen-problem

4. Specific EOM model: choice of excitation operators T, R, and the reference Φ0


Similarity transformed Hamiltonian: same spectrum as H (to prove, use eigen-state representation)

If we diagonalize it in the full many-electron basis set: we get exact energies (recover FCI)

To recover FCI, R must include up to N-electron excitations, R=R0+R1+R2+…RN

What about choice of T?


We can choose T such that the determinant Φ0 is the eigen-state of Hbar, that is:

Consider system with 2 electrons, T=T1+T2

Complete many-electron basis: {Φ0,Φia,Φij

ab}

E0 X X

0 X X

0 X X

Φ0 Φ1 Φ2

Φ0 Φ

1 Φ2

These are just regular CCSD equations!

Φex={Φia,Φij

ab}


Coupled-cluster equations

For N-electrons, T=T1+T2+ … TN and Φex={Φia,Φij

ab……..}

Φ0 is the eigenstate of Hbar and E0 is the exact energy

If we take only T1+T2, then Φ0 is not the eigen-state of Hbar operator, but is the eigenstate in the subspace of single and double excitations

Thus, the Schroedinger equation is satisfied in the subspace – connection to the “projection approach”


Conclusions:

1. CC equations can be formulated as the eigen-problem of the similarity transformed Hamiltonian.

2. Stationary properties (bi-variational principle for a non-Hermitian Hamiltonian).

3. Can recover exact energy.

4. Natural extension to the excited states (higher roots of Hbar).


EOM excited states: higher roots of Hbar:

E0 X X

0 X X

0 X X

Φ0 Φ1 Φ2

Φ0 Φ

1 Φ2


E0 X X

0 X X

0 X X

Φ0 Φ1 Φ2

Φ0 Φ

1 Φ2

EOM models: Choice of T

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

2. Amplitude equations. If T satisfies CC equations,

- size extensivity- compact wf-s, e.g., Ψexact=Φ0

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


EOM-IP: Ψ(N) =R(-1)Ψ0(N+1)

Ψ0 Ψi Ψija

EOM-EE: Ψ(Ms=0) =R(Ms=0)Ψ0(Ms=0)

Ψ0 Ψia Ψij

ab

EOM-EA: Ψ(N) =R(+1)Ψ0(N-1)

Ψ0 Ψa Ψiab

Ψ0 Ψia

EOM-SF: Ψ(Ms=0)=R(Ms=-1)Ψ0(Ms=1)

EOM MODELS: CHOICE OF R and Φ0


Size-intensivity: 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-CCSDBe(1S)Ne(1S) 0.0 0.0 0.0Be(3P)Ne(1S) 6.67 2.86 2.86Be(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) Tuesday, November 25, 2014

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). Tuesday, November 25, 2014

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).


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