Most molecules have bound higher energy excited electronic states. These states may be described by promotion of one or more electrons to higher energy orbitals. Electromagnetic energy of appropriate wavelength can cause the ground state electronic structure of a molecule to change into such state. As the rearrangement of electrons is much faster than the motion of nuclei, the nuclear configuration does not change significantly during the energy absorption process. Thus, the absorption spectrum of molecules is characterized by vertical excitation energies. Upon (usually rapid) relaxation, the nuclei adopt a new optimum geometry that is at equilibrium with the excited state electronic wavefunction. The energy difference between the relaxed excited state energy and the ground state energy is called the adiabatic excitation energy (which is measure in emission spectra). The actual absorption spectra of molecules in the gas phase are complex due to non-vertical transitions from the lowest vibrational level of the ground elecronic state into several vibrational levels of the excited electronic state. The spectra of molecules in solvents that interact strongly with the chromophore are broad and often featureless.
Meaningful excited state calculations can be difficult to carry out. There are special cases where a simple method like Configuration Interaction Singles (CIS) will give useful answers (see below). In general, however, one must be aware of pitfalls such as:
The simple CIS approach is accurate in certain special cases, in particular for so-called charge transfer transitions as discussed in class. In the CIS approach we use orbitals of the Hartree-Fock solution to generate all singly excited determinants of the configuration interaction expansion. This treatment can be thought of as the Hartree-Fock method for excited states. It allows one to simultaneously solve for a large number of excited states and to optimize the geometry of any (desired) selected state. Both spin singlet and spin triplet states can be generated. The CIS method has some appealing features:
The main problem with the CIS method is that it gives accurate excitation energies only for transitions that are dominated by single excitations. Otherwise, typical errors are 1 eV, which makes it difficult to assign observed spectral lines in the absence of symmetry. For low-lying electronic states, that are not charge transfer transitions, a form of density functional theory, known as time-dependent density functional theory (TDDFT), is preferred. For small molecules higher level wavefunction-based methods currently offer the best choice.
The table below compares the performance of different methods in predicting the UV spectrum of formaldehyde:
EXP CIS CIS-MP2 TDHF TDDFT CASSCF CASPT2 EOM-CCSD
1A2 (n -> Pi*) 4.07 4.48 4.58 4.35 3.92 4.62 3.91 4.04
1B2 (n -> Sg*) 7.11 8.63 6.85 8.59 6.87 6.88 7.30 7.04
Here CIS-MP2 is the CIS method with a perturbation correction for double excitations. TDHF stands for time-dependent Hartree-Fock, which is a modification of CIS to include ground state electron correlation. The last three columns refer to higher level methods. Note that the n -> pi** transition is given fairly well by CIS and the MP2 correction (CIS-MP2) improves the n -> sg* result considerably. In fact,the results of CIS-MP2 are very close to those of one particular high level method (CASSCF). For these specific transitions the EOM-CCSD treatment discussed briefly in class is the best.
The CIS method is implemented in the program Gaussian. To run the calculation, specify the keyword CIS. The number of desired excited states can be specifed as an option to the CIS keyword: CIS(NStates=8) requests the 8 lowest excited states. The CIS calculation is more resource-consuming than the Hartree-Fock calculation and calculations with large basis sets, such as aug-cc-pVTZ, may not be possible for larger molecules. Smaller basis such as 6-31+G(d) may be appropriate for larger molecules. Below is a sample output from the UV spectrum of a nucleobase uracil in Cs geometry with 6-31+G(2d,p) basis. The experimental spectrum of uracil in water shows two intense bands, centered around 257 nm and 220 nm. Based on the calculated intensities (f values are the oscillator strengths, which are the measure of intensity), these can be identified as excited state 2 and excited state 8. Notice that excited state 1 has very small intensity: this transition is nearly forbidden by orbital symmetry considerations. Symmetry considerations are especially useful in identifying transitions in highly symmetric molecules. In this case the CIS transition energies are significantly in error. The HOMO is orbital 43; the LUMO is orbital 44. Thus, the main contribution to state 2 (as determined by squaring the given coefficient) arises from excitation of an electron from the HOMO to LUMO + 5. You can look at the shape of orbitals involved using MOLDEN.
Excited State 1: Singlet-A" 5.7786 eV 214.56 nm f=0.0008
43 -> 44 0.57421
43 -> 45 -0.29806
43 -> 46 0.13312
43 -> 50 0.12401
This state for optimization and/or second-order correction.
Copying the Cisingles density for this state as the 1-particle RhoCI density.
Excited State 2: Singlet-A' 5.8742 eV 211.06 nm f=0.3761
43 -> 47 -0.10128
43 -> 49 0.66751
Excited State 3: Singlet-A" 6.5816 eV 188.38 nm f=0.0023
43 -> 44 0.35589
43 -> 45 0.48737
43 -> 46 -0.24746
43 -> 48 -0.11210
Excited State 4: Singlet-A" 6.6584 eV 186.21 nm f=0.0002
37 -> 49 0.23353
41 -> 49 0.54352
41 -> 57 -0.20644
41 -> 72 0.10528
41 -> 78 -0.13593
Excited State 5: Singlet-A' 7.0599 eV 175.62 nm f=0.0028
43 -> 47 0.66177
43 -> 52 -0.15451
Excited State 6: Singlet-A" 7.0916 eV 174.83 nm f=0.0037
39 -> 44 0.10592
43 -> 45 0.19574
43 -> 46 0.53360
43 -> 48 -0.12071
43 -> 50 -0.23089
43 -> 51 0.16163
43 -> 55 0.13265
Excited State 7: Singlet-A" 7.5672 eV 163.84 nm f=0.0108
39 -> 44 0.10410
43 -> 45 0.15085
43 -> 46 0.15656
43 -> 48 -0.23182
43 -> 50 0.54108
43 -> 56 0.16348
43 -> 62 -0.11541
Excited State 8: Singlet-A' 7.6361 eV 162.36 nm f=0.5017
39 -> 57 -0.10651
43 -> 52 0.10904
43 -> 54 -0.28671
43 -> 57 0.50535
43 -> 59 -0.19194
43 -> 61 -0.19798