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):1Sg (MS = 0) = ½sa sb½ = ½ { sa(1) sb(2) + sb(1) sa(2)1Sg** (MS = 0)= | s*a s*b | = ½ { s*a (1) s*b (2) - s*a (2) s*b (1) }3S* (MS = 1) = |sa s*a| = ½{ sa(1) s* a(2) - s* a(1) sa(2) }3S*(MS= -1) = |sb s*b| = ½{ sb(1) s* b(2) - s* b(1) sb(2) }3Su* (MS = 0) = ½ {½sas*b½ + ½sbs*a½}1Su* (MS = 0) = ½{½sas*b½ - ½sbs*a½}

2 determinants!

H2 molecule

• As the RHH bond stretches and breaks, the s and s* orbitals become degenerate, approaching the energy of a H 1s orbital

H2 molecule

• Expand out the lowest energy determinant:1Sg = ½sa sb½ = ½ ½ (sx + sy) a (sx + sy) b½

= ½ {½sx a sx b½+½sy a sy b½+½sx a sy b½+½sy a sx b½}

= ½ { HX·· + HY + HX + HY·· + HX· + HY· + HX· + HY· }

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·· ) }

H- + H+

ionicH+ + H-

ionicH + H

covalentH + H

covalent

H2 molecule

• Get limits for other determinants as RHH ∞ and plot the energy of the determinants as a function of RHH:

E (HX·) + E (HX·)

E (HX··) + E (HY)

½ [ E (HX·) + E (HY·)+E (HX··) + E (HY) ]

Described well asa single

determinant

1Sg

1Sg**

H2 molecule

• Get limits for other determinants as RHH ∞ and plot the energy of the determinants as a function of RHH:

E (HX·) + E (HX·)

E (HX··) + E (HY)

½ [ E (HX·) + E (HY·)+E (HX··) + E (HY) ]

Described well asa single

determinant

1Sg

1Sg**

Determinants have thesame symmetry and can interact

H2 molecule

• Mix the determinants to get the wavefunctions – the combination of determinants will vary with RHH:

E (HX·) + E (HX·)

E (HX··) + E (HY)

1Sg

1Sg**

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) molecularorbitals can be used to approx-imate excitation energies for excitation between two orbitals

– BUT energies of virtual orbitalsactually 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, i

a, with coefficients in this expansion, ci

a , 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 EOM-CCSD 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 single-electron 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 s orbitals which requires us to consider double excitations of the form sp s*p* and sp* s*p. Thus a better active space would be (4,4), ie 4 electrons in the s, p, p* and s* orbitals.

• Dynamic s 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 s orbitals which requires us to consider double excitations of the form sp s*p* and sp* s*p. Thus a better active space would be (4,4), 4 electrons in the s, p, p* and s* orbitals.

• Dynamic s 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 8.40

Serrano-Andres et al. J. Chem. Phys. 98, 3151 (1993)

EOM-CCSDEOM-CCSD(T)EOM-CCSDT-3

7.987.997.89

Watts et al. J. Chem. Phys. 105, 6979 (1995)

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)

8.067.887.69

8.097.867.69

Mueller et al. J. Chem. Phys. 110, 7176 (1999)

MRD-CI 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)

8.118.457.98

Finley et al. Chem. Phys. Lett. 288, 299 (1998)

MS-CASPT2 (2,11) 8.07 Molina et al. PCCP 2, 2211 (2000)

MS-CASPT2 (10,10) 7.95 Krawczyk et al. J. Chem. Phys. 119, 11614 (2003)

TD-DFT(LDA/ALDA)TD-DFT(LDA/VK)

7.558.05

van Faassen and de Boeij J. Chem. Phys. 120, 8353 (2004)

QD SC-NEVPT2QD PC-NEVPT2

8.068.10

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-NEVPT2RASSCF/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)