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
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