molecular quantum mechanicsmolecular quantum mechanics

- electron has cartesian and spin coordinates- electron has cartesian and spin coordinates• one electron functionsone electron functions

szyx ,,,elec- no spin operator in electronic hamiltonian - no spin operator in electronic hamiltonian

cartesian and spin coordinates are decoupledcartesian and spin coordinates are decoupled

sσzyxszyx ,,,,,elec

- spinfunction - spinfunction ((ss)) defines the spin state defines the spin state 2

1 21

- spatial part - spatial part (x,y,z)(x,y,z) is a molecular orbital is a molecular orbitalmax. two electrons (Pauli principle)max. two electrons (Pauli principle)

zyxzyx ,,,,, 21 zyxzyx ,,,,, 2

1

, or

, and

two spin states possible:

molecular quantum mechanicsmolecular quantum mechanics• molecular orbitalsmolecular orbitals

- linear combination of atomic orbitals- linear combination of atomic orbitals

rcr aoj

jjii

- - e.g.e.g. H H22

211121, rrrre

rrr 211 rrr 212 ;;

molecular quantum mechanicsmolecular quantum mechanics• atomic orbitalsatomic orbitals

- combination of simple spatial functions- combination of simple spatial functions

- mimic atomic s,p,d,… orbitals- mimic atomic s,p,d,… orbitals

Slater-type orbitals:Slater-type orbitals:

3

1

331,1,1

21,4

1

8i

xsisis

siedx

3

1

332,2,2

22,4

1

8i

xspisis

spiedx

3

1

352,2,2

22,4

1

128i

xspipip

spi

xexdx

gaussian-type orbitals:gaussian-type orbitals:

e.ge.g. sto-3g basisset (2D) . sto-3g basisset (2D)

i i and and ii are constants: basisset (sto-3g, 3-21g, 6-31g*, ..) are constants: basisset (sto-3g, 3-21g, 6-31g*, ..)

i

RriN

Nier

i

RriN

Nier2

molecular quantum mechanicsmolecular quantum mechanics• atomic orbitalsatomic orbitals

- combination of simple spatial functions- combination of simple spatial functions

- mimic atomic s,p,d,… orbitals- mimic atomic s,p,d,… orbitals

Slater-type orbitals:Slater-type orbitals: i

RriN

Nier

i

RriN

Nier2

3

1

331,1,1

21,4

1

8i

rsisis

siedr

3

1

332,2,2

22,4

1

8i

rspisis

spiedr

3

1

352,2,2

22,4

1

128i

rspipip

spi

xexdr

gaussian-type orbitals:gaussian-type orbitals:

e.ge.g. sto-3g basisset (3D) . sto-3g basisset (3D)

i i and and ii are constants: basisset (sto-3g, 3-21g, 6-31g*, ..) are constants: basisset (sto-3g, 3-21g, 6-31g*, ..)

molecular quantum mechanicsmolecular quantum mechanics• Hartree-Fock wavefunctionHartree-Fock wavefunction

- anti-symmetric product of molecular orbitals- anti-symmetric product of molecular orbitals

- molecular orbitals- molecular orbitals

- atomic orbitals with fixed coefficients (basisset)- atomic orbitals with fixed coefficients (basisset)

• optimization of MO coefficents optimization of MO coefficents ccjiji

- variation principle- variation principle

- find - find ccjiji that minimize the energy that minimize the energy

nmnmaane rrrrrrr 12121 ,,,

rcr aoj

jjii

0* ˆ EdH ee

molecular quantum mechanicsmolecular quantum mechanics• Hartree-Fock based methodsHartree-Fock based methods

- Hartree Fock wavefunctionHartree Fock wavefunction

- MCSCF (CI, CASSCF)MCSCF (CI, CASSCF)

- perturbation theory (MP2, MP4, CASPT2)perturbation theory (MP2, MP4, CASPT2)

• alternative methodsalternative methods- semi-empiricalsemi-empirical methods methods

- density functional theory methodsdensity functional theory methods

- high demand on computational resourceshigh demand on computational resources

no electron correlationno electron correlation

mean field approach:mean field approach:

as starting pointas starting point

dre rr

n

jre kjh

e

ij

,,,,2 2

excited state quantum chemistryexcited state quantum chemistry• Hartree-Fock approximation for ground stateHartree-Fock approximation for ground state

nn1n1n2211HF φφφφφφφφψ

• all are optimized (self-consistent field)all are optimized (self-consistent field)jic

• no static/dynamic correlationno static/dynamic correlation

2αao

j αχ nucRre

j

aojji

moi χcφ

2n electrons2n electrons

excited state quantum chemistryexcited state quantum chemistry• Complete Active Space SCFComplete Active Space SCF

• and are optimized simultaneouslyand are optimized simultaneously

• resolves (part of) static correlationresolves (part of) static correlation

k

HFkk

CAS ψCΨ

kC jic

• excited statesexcited states

HFCASΨ Coror

excited state quantum chemistryexcited state quantum chemistry• simple (but incorrect) CAS expansionsimple (but incorrect) CAS expansion

1nn112nn111CAS φφφφCφφφφCΨ

0C,1C 21 • ground state Sground state S00

1C,0C 21 • excited state Sexcited state S11

• optimize 2optimize 2ndnd root root

electronic transitions in QM/MM electronic transitions in QM/MM • diabaticdiabatic surface hopping surface hopping

ψψΨCAS21 CC

- t- t11::

012 t2

t1 CC

21 ψ1ψ0Ψ

- t- t22: : 21 ψ1ψ0Ψ

- t- t33: : 21 ψ0ψ1Ψ

- t- t33: : 21 ψ1ψ0Ψ

- t- t44: : 21 ψ1ψ0Ψ

- swap electronic states- swap electronic states

123 t2

t1 CC

034 t2

t1 CC