l3-methods for excited states - light and molecules€¦ · for excited states, the important...

MethodsforexcitedstatecalculationsMario BarbattiA*Midex Chair Professormario.barbatti@univ‐amu.fr

Aix Marseille Université, Institut de Chimie Radicalaire

L3

L I G H T A N DM O L E C U L E S

2

0e i iH E


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Configuration Interaction (CI)


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1 1 2 1 1

1 2 2 2 21 2 1 2

1 2

1, , ,!

N

NN N

N N N N

N

x x x

x x xx x x

x x x

Slater determinant for N electrons

0 1 2

1 2

1 2

rr

a b N

a b N

rsr sab N

HF

S

D

Single substitution

Double substitution


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0 0 S DS D

c c S c D

Variational Principle

1E E

EHc Sc

ˆij i j

ij i j

H H

S


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CI wavefunctionCFS CFS CFS

i i i j iji

c

EHc c

The CI problem is to solve

• Often, CI expansion is done in terms of CFS’s (configuration state functions).

• CFS’s are linear combinations of Slater Determinants that are orthogonal and eigenvectors of spin and angular momentum operators.

• Basis of CFS’s are built with GUGA (graphical unitary group approach).


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EHc c2 2

1 1

0 0

Energy Wavefunction

EEE

ccc

Ground state

Excited states


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• Clear hierarchy of levels:

CIS < CID < CISD < … < Full CI

• CI is time consuming:

CIS < CID < CISD < CISD+Q

Davidson correction


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Coupled Cluster (CC)


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1 1 2 2 n n

TCC e HF

T t t t

• Sneskov and Christiansen, WIREs 2, 566 (2011)


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1 1 2 2 n n

TCC e HF

T t t t

i it

amplitudes

Excitation operator ith‐order


ij

ab

2abij

ij

ab


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1 1 2 2 n n

TCC e HF

T t t t


• Choose a truncation level• Insert this Ansatz into TDSE• Get a set of nonlinear equations for the amplitudes and ground‐state energy

i

T Ti

T TCC

e t e He HF

E HF e He HF


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1 1 2 2 n n

TCC e HF

T t t t


CCS < CCSD < CCSDT < ···

Truncation produces a well defined hierarchy of methods:

CC3, CCSD(T)With some exotic flavors


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i

T Tie t e He HF

For excited states, the important quantity is the CC Jacobian matrix A:

i

i i

i

eA

t

Response Theory shows that:• Excited‐state energies are eigenvalues of A• Contribution R from each determinant is the eigenvector

The problem is to solve AR R


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i i

i i i i

i i

e eA A

t t

Jacobian is not a symmetric matrix!

1

AR RLA LLR

=====================================

Left and right eigenvalues converged to the same Results within 0.42E-07 a.u.


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CCS < CCSD < CC3 < CCSD(T) < CCSDT < ···

CC is time‐consuming…

CC2CIS(D∞)


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CC doesn’t work well for degenerated excited states

Non‐symmetric Jacobian→ eigenvalues (energies) may be imaginary

CCS < CIS(D∞ ) < CC2 < CCSD < CC3 < CCSD(T) < CCSDT < ···

CIS D A

†1ADC 2 CIS D CIS D2

A A A

Build a symmetric Jacobian:

• Hättig, Köhn, J Chem Phys 117, 6939 (2002)


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CCS < CIS(D∞ ) < CC2 < CCSD < CC3 < CCSD(T) < CCSDT < ···

ADC(2)

ADC(2): Algebraic Diagrammatic Construction scheme up to second order

ADC(2) is not strictly a CC method

It is a excited‐state propagator for ground‐state MP2 (Schirmer1982)

These excited states are equivalent to the symmetrized CIS(D∞)

Trofimov, Krivdina, Weller, Schirmer, Chem Phys 329, 1 (2006)


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Time-dependent Density Functional Theory (TDDFT)


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Which frequency of an external force will induce resonance in an oscillator?


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ˆ ' ' 'O t t t t dt

susceptibility perturbationexpected value

' F ' 'x t t t t dt

susceptibility forceposition

, , ', ' ', ' ' 'appt t t v t dt d

r r r r r

susceptibility perturbationdensity variation


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Linear response

, , ', ' ', ' ' 'appt t t v t dt d

r r r r r

Fourier space

, , ', ', 'app appv d v r r r r r


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Non‐interacting Kohn‐Sham

s s sv

0 0

v v

v vs eN app H XC

H XC

v V v

In terms of KS orbitals , occupations f, orbital energies , and XC functional AXC, the matrix elements of and K are:


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Runge‐Gross Theorem

s s sv v

1 1 Ks

sv

Reordering orbitals in terms of occupation:

1* *

A BI 0B A0 I

Energies are poles of the susceptibility (system resonance).Therefore, they can be obtained by solving:

* *

A B I 0B A 0 I

X XY Y


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Multiconfigurationalself-consistent field (MCSCF)


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i ii

c CI combination of Slater determinants

1 2k Slater determinants combination of spin‐orbitals

2l l r Spin‐orbital is product of molecular orbital and spin function

n m mm

C Molecular orbital combination of atomic orbitals (basis functions)

,f C cTo solve CI wavefunction, two sets of coefficients must be determined, C and c.

Two strategies:

CI (configuration interaction)1. First, get molecular orbitals (find C)2. Then, solve CI (find c)

MCSCF (multiconfigurational self‐consistent field)1. Get C and c simultaneously


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MCSCF in a small subspace of molecular orbitals

For instance:

C 1sC 1sCC

CH

CH

CH

CH

CAS(2,2)


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But is can be much more complicated

• Sellner, Barbatti, Müller, Domcke, Lischka, Mol Phys 111, 2439 (2013)


Non-dynamical electron correlation?


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• The non‐relativistic Born‐Oppenheimer WF is:

1 2 1 2 1 2 1 2, , , ; , , , , , , ,N N N Ns s s s s s

r r r r r r

Electronic wavefunction Spatial part Spin‐part

Pauli Principle: AA S

S A=

x

xor

The spatial part of the WF should belong to a representation of the permutation group SN in the same ways that it belongs to a representation of the point group.


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1 2 3 2 1 3 3 2 1 2 3 1

1 2 3 2 1 3 3 2 1 2 3 1

C

• The wrong permutation symmetry of the Slater determinant is the origin of the non‐dynamical correlation problem.

• 3‐e WF with correct permutation symmetry (S3 group):

• 3‐e Slater Determinant :

1 2 3 2 3 1 3 1 2

3 2 1 2 1 3 1 3 2

13!

1 2 3


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Multireference Methods (MR)


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Energy

Time

SR

ia SR

ja SR

SR

i ja a SR

i ja a SR

MR

MR

Dynamic Electron Correlation

Static EC

Certain states cannot be described by a single reference. They need multireference methods


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Usual way is to compute dynamics correlation on the top of a multiconfigurational WF:

CASSCF + CI → MRCI

CASSCF + CC → MRCC

CASSCF + PT → MRPT


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0 0 0 0ˆ ˆ

SRCI s ds d

c c S c D

Single‐reference CI: expansion based on one single function 0

0ˆ ˆ

MRCI n n sn n dnn s n d n

c c S c D

Multi‐reference CI: expansion based on small set of functions n


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CAS(2,2) as reference space for MRCI

C 1sC 1sCC

CH

CH

CH

CH

CAS(2,2)

1 2 3


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MR‐CISD(2,2)

C 1sC 1s

CC

CH

CH

CH

CH

CISD +

Frozen C 1sC 1s

CC

CH

CH

CH

CH

CISD +

Frozen C 1sC 1s

CC

CH

CH

CH

CH

CISD

Frozen


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0 020 0 0

CAS CSF CSF CASi I I iCASPT CAS CAS

i iI i I

V H V HE H

E

0H H V

00

CAS CASI I IH E

Energy to the second order

Perturbation approach


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Overview of methods for excited states


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• Single‐reference • NAC not available

Available for EOM‐CC: Tajti and Szalay, J Chem Phys 131, 124104 (2009)

• Relatively fast (CC2)• Good vertical energies

Schreiber et al., J Chem Phys 128, 134110 (2008)


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Tip: crossing between excited states are well described• Barbatti et al., Chem Phys 375, 26 (2010)

• Problems with charge‐transfer and Rydberg statesDreuw and Head‐Gordon, Chem Rev 105, 4009 (2005)

• Problems with Kohn‐Sham convergence at multireference regionsPerdew et al., JCTC 5, 902 (2009)

• Conical intersections have wrong dimensionality

• Very fast• Gradients• Good vertical energies

Silva‐Junior et al., J Chem Phys 129, 104103 (2008)Jacquemin et al., JCTC 5, 2420 (2009)


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TDDFTB Same advantages and disadvantages than TDDFT, but with poorer results Very fast (semi‐empirical)

• Elstner, Theor Chem Acc 116, 316 (2006)

DFT/MRCI Fast and good results (KS convergence still a problem at MR regions) No gradients available

• Grimme and Waletzke, J Chem Phys 111, 5645 (1999)

Single‐determinant Kohn‐ShamLarge (condensed‐phase) systems can be treatedQuality of simulations is unclear

• Craig et al., Phys Rev Lett 95, 163001 (2005)


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Tip: Combination of CAS + GVB‐PP stabilizes the dynamics• Sellner et al., J Chem Phys 131, 024312 (2009)

• Bad description of ionic statesAngeli, J Comput Chem 30, 1319 (2009)

• Too expensive• Too sensitive to active‐space choice

Szymczak et al., Int J Quantum Chem 111, 3307 (2011)

• Clear and intuitive• Multireference / multiconfigurational• Gradients, nonadiabatic couplings (NAC) available


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Tip: Beware of good results for wrong reason! It tends to overes mate ionic states ↑Conventional zero‐order Hamiltonian tends to underes mate energies ↓

• Ghigo et al., Chem Phys Lett 396, 142 (2004)

• Sensitive to active‐space choice• Too expensive for dynamics• Analytic gradients and NACs not available

• Much faster than MRCI• Muticonfigurational• Good vertical excitations

Schreiber et al., J Chem Phys 128, 134110 (2008)• The best option for potential‐surface calculation


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Tip: MRCI available in combination with AM1, PM3, and OM2• Granucci et al., J Chem Phys 114, 10608 (2001)• Koslowski et al. J Comput Chem 24, 714 (2003)

• Excited‐state surfaces are not fully reliable• Parameters are neither complete nor fully transferable

• Very, very fast• Good vertical energies at OMx levels

Silva‐Junior and Thiel, JCTC 6, 1546 (2010)


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• There are many different methods available to compute excited states.• They differ in terms of accuracy and computational costs.• To be able to determine which one is the most adequate for each case is essential.

l3-methods for excited states - light and molecules€¦ · for excited states, the important...

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