Breaking Bonds: Consider the H2 Molecule
We can draw out possible electron configurations (configuration state
functions/determinants) with which to represent the H2 wavefunction (ie these
are NOT wavefunctions):
1S
g
1S
g**
3S*
(MS = 0) = = ½ { (1) (2) + (1) (2)
(MS = 0)= | ** |= ½ { *(1) * (2) - *(2) * (1) }
(MS = 1) = | *| = ½{ (1) * (2) - * (1) (2) }
3S*(M = -1) = | *| = ½{ (1) * (2) S
3S * (M = 0) = ½ {*+ *}
u
S
1S * (M = 0) = ½{*- *}
u
S
* (1) (2) }
2 determinants!
H2 molecule
• As the RHH bond stretches and breaks, the  and *
orbitals become degenerate, approaching the energy of
a H 1s orbital
H2 molecule
• Expand out the lowest energy determinant:
1S
g
= = ½ (sx + sy) (sx + sy) 
= ½ {sx sx +sy sy +sx sy +sy sx }
= ½ { HX+ HY + HX + HY + HX+ HY+ HX+ HY }
H- + H+
ionic
H+ + Hionic
H+H
covalent
H+H
covalent
E(1Sg)  ½ [E (HX) + E (HX) + E (HX) + E ( HX ) ]
ie the average of the covalent energy { E (HX) + E (HX) } and
the ionic energy { E (HX) + E ( HX ) }
H2 molecule
• Get limits for other determinants as RHH   and plot the
energy of the determinants as a function of RHH:
1S
g**
E (HX) + E (HY)
½ [ E (HX) + E (HY)+E (HX) + E (HY) ]
1S
Described well as
a single
determinant
E (HX) + E (HX)
g
H2 molecule
• Get limits for other determinants as RHH   and plot the
energy of the determinants as a function of RHH:
1S
g**
E (HX) + E (HY)
½ [ E (HX) + E (HY)+E (HX) + E (HY) ]
1S
Described well as
a single
determinant
E (HX) + E (HX)
g
Determinants have the
same symmetry
and can interact
H2 molecule
• Mix the determinants to get the wavefunctions – the
combination of determinants will vary with RHH:
1S
g**
E (HX) + E (HY)
E (HX) + E (HX)
1S
g
Now we dissociate to the right things…
H2 molecule
• At intermediate RHH distances, the 1Sg and 1Sg ** valence
determinants interact: multiconfiguration problem
• The triplet dissociation is well behaved...
• singlet-triplet instability – the spin-paired wave function is
unstable with respect to relaxation of the spin symmetry ie the
energies of the individual singlet and triplet determinants cross
• This instability occurs at a threshold when the exchange interaction
between the electrons involved exceeds their orbital energy difference
• That is, te singlet-triplet instability arises from competition between
spin-pairing in a bond and spin localization in separated atoms.
• Somewhat unfortunately, the singlet-triplet instability is ubiquitous
when studying breaking bonds and can be much worse for multiple
bonds…
Excited States
• Koopman’s Theorem: the molecular orbital energy
approximates energy required to ionize an electron from
that orbital.
– Energies of the occupied and
unoccupied (virtual) molecular
orbitals can be used to approximate excitation energies for
excitation between two orbitals
– BUT energies of virtual orbitals
actually correspond to states
with N electrons whereas
electron in virtual orbital
should only see N-1 electrons
Methods for Excited States
• Ground State Methods
– a ground state method can be used to calculate the lowest
energy state for each possible spin multiplicity (2S+1)
– a ground state method can be used to calculate the lowest
energy state for each possible irreducible representation of the
wavefunction in the molecular point group (spatial symmetry)
• Single Reference Methods for Excited States
– ONLY if the excited state is dominated by a single determinant
or if the multi-configurational excited states (eg open shell
singlets) are correctly described by single reference methods
provided their wavefunctions are dominated by single-electron
excitations
– Closed shell species at equilibrium geometries
– Some doublet radicals
– Some triplet diradicals
Methods for Excited States
• Configuration Interaction Single Excitations (CIS)
– The CIS wavefunction starts with an optimized HF reference
wavefunction
– All the “excited” Slater determinants representing single electron
excitations from the O occupied orbitals to the V virtual
(unoccupied) orbitals are constructed and the electronic
wavefunction is expanded as a linear combination of these
determinants, ia, with coefficients in this expansion, cia , are
determined variationally
– Diagonalizing the matrix representation of the Hamiltonian in the
space of singly excited determinants yields eigenvalues
corresponding to the energies of the ground and excited states
and eigenvectors corresponding to the ground and excited
electronic state wavefunctions.
CIS: Properties and Limitations
– Can be applied to larger molecules
– The CIS wavefunction is variational ie excited state energies are
upper bounds to the exact energies
– The excited state wavefunctions are orthogonal to the ground
state wavefunction
– CIS is size consistent
– It is possible to obtain pure singlet and pure triples states for
closed shell molecules.
– The CIS excited state wavefunctions are “well-defined”
– The CIS energy is analytically differentiable  efficient
optimizations
CIS: Properties and Limitations
– Does not explicitly include correlation through the ground state
wavefunction
– In general excitation energies at CIS are too large by 0.5-2 eV
compared with experiment.
– The “singly excited” HF determinants are poor first-order
estimates of the true excitation energies (since the orbitals are
not allowed to relax on excitation).
– Transition moments are not accurate (they do not sum to the
number of electrons!) so at best they provide a qualitative guide.
TDHF – time dependent HF (Dirac 1930)
• An approximation to the exact time-dependent Schrödinger equation
and assume that the system can be represented by a single Slater
determinant composed of time-dependent single-particle
wavefunctions, (r,t).
• Implementation is as the linear response
• We get time-dependent HF equations using a time-dependent
Hamiltonian: H(r, t)= H(r) + V(r,t), eg in a time-dependent electric
field, V(r,t).
At t=0 start with single Slater determinant 0(r).
Apply very small time-dependent perturbation
This causes a very small change in the orbitals of the Slater determinant.
The TDHF equations calculate the first order response (the linear response) of
the orbitals and the Fock operator to the applied perturbation.
– This response is characterized by excitation of electrons from orbital i to orbital a
within the Slater determinant and the linear response of the Coulomb and
exchange operators to V(r,t).
– The excited states are effectively resonances in the linear response.
–
–
–
–
TDHF Properties and Limitations
• The CIS method is contained “within” the TDHF method and TDHF
exhibits similar properties to CIS
• Can be applied to larger molecules
• Yields excitation energies and transition vectors.
• It contains not only “singly excited” states but “singly de-excited
states”
• Gives better transition moments than CIS (they sum to N)
• Analytic energy derivatives are accessible  efficient optimizations.
• Does not explicitly include correlation through the ground state
wavefunction
• Poor at predicting triplet spectra because the HF reference state can
lead to triplet instabilities (which are not a problem in CIS).
• Excitation energies are only slightly smaller than at CIS and are still
overestimates
• Computational cost is about twice CIS and this is usually not justified
EOM-CC Equations-of-Motion Coupled
Cluster Methods
• Linear response versions of Coupled Cluster theory
• Linear excitation mixes in excited state character into the
wavefunction which can then be analysed.
EOM-CCSD (and CISD) scales as N6.
Limited to fairly small molecule.
truncated versions are more accurate than similarly truncated CI
EOM-CC methods are rigorously size-extensive.
Analytic gradients are possible  optimizations and properties.
Depending on the level of truncation in the CC expansion, EOM-CC
methods can yield very accurate results: 0.1-0.3 eV accuracy in
excitation energies.
• A T1 diagnostic > 0.02 casts suspicion on the applicability of single
reference methods.
•
•
•
•
•
•
TDDFT – time dependent DFT
• TDDFT calculates linear time-dependent response of the electron
density to a small, time-dependent perturbation, V(r,t). The
formalism is equivalent to the TDHF equations and excitation
energies and transition vectors can be obtained similarly.
• The Tamm-Dancoff approximation (TDA) is equivalent to the CIS
approximation to TDHF but TDA/TDDFT is a very good
approximation to TDDFT (better than CIS to TDHF) presumably
because electron correlation was included in the ground state
electron density.
• TDDFT is more resistant to triplet instabilities than TDHF.
• The B3LYP and PBE functionals are probably the most widely used
functionals within TDDFT.
TDDFT – Properties and Limitations
• Electron correlation is included in the ground state wavefunction
• Can be applied to larger systems
• TDDFT results often very sensitive to the functional: need to
benchmark.
• Typical TDDFT errors are 0.1-0.5 eV for electronic excitation
energies involving valence states (almost comparable with EOMCCSD or CASPT2!) however, to reach this accuracy a large set of
virtual orbitals must be used in the Kohn-Sham equations, ie a large
basis set.
• TDDFT is so accurate because (in contrast to TDHF) the Kohn-Sham
orbital energies are usually excellent approximations for excitation
energies.
• Since the derivation of TDDFT is analogous to TDHF it is variational
within the “model chemistry” of the functional used.
TDDFT – Properties and Limitations
•
•
•
•
TDDFT is size-consistent
It gives better oscillator strengths than CIS
Analytic energy derivatives are accessible  efficient optimizations
TDDFT does not describe Rydberg states correctly, valence states
involving extended p systems, doubly excited states and charge
transfer states. In these cases the errors in excitation energies can
be 1-2 eV.
• These problems arise because the long range behaviour of the
exchange-correlation terms is incorrect (they decay faster than 1/r).
• States with double excitation character cannot be treated within the
TD formalism (either TDHF or TDDFT) because the linear response
formalism only contains single excitations.
Case Study: torsional motion in ethylene
Krylov, Acc. Chem. Res. 39, 83 (2006).
Case Study: torsional motion in ethylene
• Around equilibrium, the ground-state wavefunction of ethylene (the
N-state) is dominated by the p2 configuration.
• As the CC bond twists a degeneracy between p and p * develops
along the torsional coordinate and the importance of the (p *)2
configuration increases until, at the torsional barrier, p and p * are
exactly degenerate; wavefunction must include both configurations
with equal weights.
• NB Even when the second configuration is explicitly present in a
wave function (e.g., as in the CCSD or CISD models), it is not treated
on the same footing as the reference configuration, p 2.
• The singlet and triplet p p * states (V and T) are formally singleelectron excitations and are well-described by single reference
excited state models like the EOM-CC methods
• The Z-state is formally a doubly excited state and single reference
models will not treat it accurately.
Case Study: torsional motion in ethylene
• minimal active space is 2 electrons placed in 2 orbitals, p, p* (2,2)
• The full valence space is 12 electrons in 12 orbitals so, using 2 active
orbitals, we have 10 inactive orbitals (with 10 electrons) describing
the C–C and C–H bonds.
• Unfortunately the picture is a little more complicated, we have
missed dynamic correlation.
• In ethylene there is dynamic polarization of the  orbitals which
requires us to consider double excitations of the form p  *p*
and p*  *p. Thus a better active space would be (4,4), ie 4
electrons in the , p, p* and * orbitals.
• Dynamic  polarization leads to contraction of the p atomic orbitals.
• Dynamic correlation of the p electrons.
• There are a number of Rydberg states very close in energy to the V
state, for which dynamic correlation energy is lower than in the V
state (which has valence character)…
• Test N-V wrt vertical transition energy
Case Study: torsional motion in ethylene
• A minimal active space to describe the torsional motion involves 2
electrons placed in 2 orbitals, p, p*, denoted (2,2)
• The full valence space is 12 electrons in 12 orbitals so, using 2 active
orbitals, we have 10 inactive orbitals (with 10 electrons) describing
the C–C and C–H bonds.
• In ethylene there is dynamic polarization of the  orbitals which
requires us to consider double excitations of the form p  *p*
and p*  *p. Thus a better active space would be (4,4), 4
electrons in the , p, p* and * orbitals.
• Dynamic  polarization leads to contraction of the p atomic orbitals.
• Dynamic correlation of the p electrons.
• There are a number of Rydberg states very close in energy to the V
state, for which dynamic correlation energy is lower than in the V
state (which has valence character)… need to include
valence/Rydberg mixing…
Case Study: torsional motion in ethylene
Method (Electrons, Active Orbitals)
Energy (eV)
Reference
MR-CISD
7.96
McMurchie and Davidson J Chem Phys, 67, 5613 (1977).
INO/MR-CISD
8.01
Brooks and Shaefer Chem Phys, 68, 4839 (1978).
MC SCF
7.97-8.09
Sunil et al. Chem Phys, 88, 55 (1984).
MR-CISD
7.94
Lindh and Roos Int. J. Quantum Chem. 35, 813 (1989)
CASSCF (2,11)
CASPT2 (2,11)
8.20
Serrano-Andres et al. J. Chem. Phys. 98, 3151 (1993)
EOM-CCSD
EOM-CCSD(T)
EOM-CCSDT-3
MR-AQCC/CAS(2,2)
MR-AQCC/CAS(6,6)
MR-AQCC/CAS(12,12)
MR-CISD+Q/CAS(2,3)
MR-CISD+Q/CAS(6,7)
MR-CISD+Q/CAS(12,13)
8.40
7.98
7.99
7.89
8.06
7.88
7.69
Watts et al. J. Chem. Phys. 105, 6979 (1995)
Mueller et al. J. Chem. Phys. 110, 7176 (1999)
MRD-CI
8.09
7.86
7.69
7.90-7.95
Krebs and Buenker J. Chem. Phys. 106, 7208 (1997)
CAS(2,3)-SDCI
7.99
Pérez-Casany et al. Chem. Phys. Lett. 295, 181 (1998)
CASSCF (12,13)
CASPT2 (12,13)
MS-CASPT2 (12,13)
MS-CASPT2 (2,11)
8.11
8.45
7.98
8.07
Finley et al. Chem. Phys. Lett. 288, 299 (1998)
MS-CASPT2 (10,10)
7.95
Krawczyk et al. J. Chem. Phys. 119, 11614 (2003)
TD-DFT(LDA/ALDA)
TD-DFT(LDA/VK)
7.55
8.05
van Faassen and de Boeij J. Chem. Phys. 120, 8353 (2004)
QD SC-NEVPT2
QD PC-NEVPT2
8.06
8.10
Angeli et al. J. Chem. Phys, 121, 4043 (2004)
MR-CISD1Q/SA-3-RDP (2,2)
7.80
M. Barbatti, J. Paier and H. Lischka J. Chem. Phys. 121, 11614 (2004)
RASSCF/PC-NEVPT2
RASSCF/CASPT2
~7.75
~7.80
Angeli J. Comp. Chem. in press (2008)
Experiment
~8.0
P. D. Foo and K. K. Innes, J. Chem. Phys. 60, 4582 (1974)
Molina et al. PCCP 2, 2211 (2000)