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