The accurate prediction of electronic transition energies and intensities remains a challenging area of quantum chemistry. Many of the early successes of MO theory involved qualitative interpreta- tions of electronic spectra. The development of the Hu¨ckel model292,293 made semiquantitative calculations for conjugated organic molecules possible for the first time. The crystal field model (see Chapter 2.35) was successfully applied to calculated–d transitions in TM complexes.
Much of our qualitative understanding of electronic transitions and the language we use to describe them derives from the independent particle approximation (IPA). Typically, one interprets features in absorption spectra of TM complexes as d–d, MLCT, LMCT, or LLCT transitions, etc., in an attempt to describe these processes in terms of the simple MO picture. However, such a description is approximate and the actual electron excitation process is more complex. Within the Born–Oppenheimer approximation,1 the interaction between specific UV or visible light and matter leads to transitions between quantum states that belong to separate potential energy surfaces. In general, the absorption of a photon takes a state characterized by distinct electronic, vibrational, and rotational quantum numbers to another state with a different set of quantum numbers. The energy required for transitions between different electronic states is typically much larger than energy differences between vibrational and rotational states so typically observed low-resolution spectra are often character- ized by absorption peaks that grossly represent the electronic part of the transition, with the underlying vibrational and rotational features unresolved. This is especially true for TM species in solution where the fine structure of absorption bands is rarely resolved.
Here we discuss the most common ways to calculate electronic spectra. Note that these methods can be used not only with semiempirical methods, but with other quantum mechanical methods as well.
2.38.4.1 Single-transition Approximation
Let’s assume that the MO model is a valid first-order approximation and that the radiation field causes a single electron to be promoted from the occupied MO i to the MO a, which is not occupied in the ground electronic state. Furthermore, the MOs in the ground and excited states are the same (thefrozen orbitalapproximation)1and there is no interaction between excited states so that an excited state can be represented by one excitation,i!a. This is the so-called single- transition approximation (STA).294
In the case that the ground state is a closed-shell singlet, excitation of an electron fromiwith MO eigenvalue"i toawith MO eigenvalue"agives rise to a singlet and a triplet configuration (the case where bothiandaare degenerate orbitals is not considered here). In STA, the energy corresponding to a singlet–singlet transition is given by:294
!singleti!aẳ"a"iJiaỵ2Kia ð43ị whereJiaẳ(ii|aa) andKiaẳ(ia|ai) are the molecular two-center Coulomb and exchange integrals respectively.1The energy of the corresponding singlet–triplet transition is:
!tripleti!aẳ"a"iJia ð44ị
Similar expressions exist for electronic states of other multiplicities.70Note that, unlike the Hu¨ckel method,292,293the STA model does not predict that the transition energy is equal to a difference of MO eigenvalues,"a"i. Rather, it allows for a certain amount of interaction between electrons and correctly predicts unequal energies for electronic transitions that differ in the multiplicity.
Oscillator strengths,fI, of electronic transitions in the dipole length and velocity formulations are:2 fIrẳ2
3!Ih0jrjIi2 ð45ị
fIrẳ 2
3!Ih0jrjIi2 ð46ị
Semiempirical SCF MO Methods, Electronic Spectra, and Configurational Interaction 479
where r and r are the dipole length and velocity operators respectively. In STA, transition moments are:
h0jrjIiSTAẳ ffiffiffi p2
hijrjai ð47ị
h0jrjIiSTAẳ ffiffiffi 2
p hijrjai ð48ị
for the I-th electronic transition corresponding to the i! aexcitation.
Although the STA model may be qualitatively appealing, it does not provide quantitatively satisfactory predictions. The principal reason for this is the fact that electronic transitions can rarely be described as just single excitations. There is a better theoretical approach than STA, which does not discard the intuitively attractive one-electron picture. This method is known as configuration inter- action with single excitations (CIS or SECI) or the Tamm–Dancoff approximation (TDA).2,3,72,295
2.38.4.2 Configuration Interaction with Single Excitations
All possible Slater determinants can be described by reference to the HF determinant,0. So, the exact wave function for any electronic state of the system can be written as
IẳC0;I0 ỵ X
i;a
Cia;Iai þ X
i<j a<b
Ciajb;Iabij þ X
i<j<k a<b<c
Ciajbkc;Iabcijk ỵ ð49ị
where ai are singly excited determinants, abij are doubly excited determinants, abcijk are triply excited determinants, etc., and indexesi,j,kanda,b,crefer to occupied and unoccupied MOs of the HF ground state respectively.1Multiply excited states are of less interest in electronic spectros- copy than singly excited states since the electronic transitions from the ground state to multiply excited states are forbidden. However, such states can interact and mix with singly excited states, thereby affecting their properties.
An infinite set of N-electron determinants, {ai, abij, abcijk,. . .}, is a complete set for the expansion of any N-electron wave function, and the energies of the ground and excited states of the system are the eigenvalues of the Hamiltonian matrix. This procedure is called configura- tion interaction (CI).1–3Practical applications of this method to large molecular systems are often restricted to the CI expansion with single excitations (CIS) or to the CI expansion with single and double excitations (CISD).2
In the CIS method, each excited state is approximated by a linear combination of frozen-orbital single-electron excitations,ai. First, the set of single-electron excitations is used to construct the CIS matrix, A. For spin singlet-singlet transitions,
Aiajbẳ h1aijHj1bji h10jHj10iijabẳ ð"a"iịijabỵ2ðaijjbị ðabjjiị ð50ị Then, the eigenvalue problem is solved:
ACIẳ!ICI ð51ị
The eigenvalues,!I, are the transition energies. Since only single excitations are used in the CIS approximation, this approach fails in the description of excited states with significant contribu- tions from doubly and triply excited determinants, abij ,abcijk.
Transition moments in the dipole length and velocity formulations are given by:
h0jrjIiCISẳ ffiffiffi 2 p X
i;a
Cia;Ihijrjai ð52ị
h0jrjIiCISẳ ffiffiffi p2X
i;a
Cia;Ihijrjai ð53ị
The CIS approximation is used in a majority of semiempirical SCF methods to calculate energies and oscillator strengths of electronic transitions (CNDO/S, INDO/S, CINDO-E/S, NDDO/MC, NDDO-G).
2.38.4.3 Direct Methods
When applied to spectroscopy, the response methods, also called Green’s function methods or Liouville methods, yield the energies and the intensities of electronic transitions directly.1,2 Starting with the generalized Schro¨dinger equation,
HjIi ẳEIjIi ð54ị
let’s define an excitation operatorQIthat when operating on the ground state |0i, generates an excited state |Ii:
QIj0i ẳ jIi ð55ị
If the Liouvillian operatorLI is defined from the commutator [H,QI]:
LI ẳ ẵHQI ẳHQIQIH ð56ị LIj0i ẳHQIj0iQIHj0i ẳHjIiQIE0j0i ẳEIjIiE0QIj0i ẳEIjIiE0jIi ẳ!IjIi ð57ị the transition energies!I appear naturally as a consequence of these calculations. Note that 0 occurs on the left side ofEquation (57), butIon the right. This points out one of the difficulties of this method: it is not easy to extract information on the individual excited states. On the good side, this methodology generally provides not only accurate transition energies, but also reliable transition intensities.2
2.38.4.3.1 Random-phase approximation
The simplest of the direct methods is the RPA.2,69,72,296–302
This approximation uses the SCF wave function for0and the excitation operator QI,
QIði!aị ẳ ẵðaỵaai ỵ aỵaaiị ỵ ðaaaỵi ỵ aaaỵiị= ffiffiffi 2
p ð58ị
where aþa is the creation operator, which creates an electron in spin orbital a, and aa is the annihilationoperator, which removes an electron from a:1
aỵaj k::: li ẳ j a k::: li ð59ị aaj a k::: li ẳ j k::: li ð60ị RPA with real orbitals can be expressed as a non-Hermitian eigenvalue problem,
A B B A
! Z Y !
ẳE 1 0
0 1
! Z Y !
ð61ị
whereAis the CIS matrix,E is the diagonal matrix of transition energies, and the (Z,Y) vector denotes the RPA eigenvector solution ofEquation (61). The matrix elements forB are
Biajbẳ 2ðiajjbị ðajjbiị ð62ị
for singlet–singlet transitions. TheBmatrix represents interaction between the ground state and doubly excited states. The HF ground state is used to evaluate theA andBmatrices. However, the PRA ground state will consist of the HF ground state plus doubly excited states, i.e., the effects of configuration interaction are, to some extent, taken into account. This, as expected, is an improvement over CIS.
In RPA, transition moments in the dipole length and in the velocity formulations are:
h0jrjIiRPAẳ ffiffiffi p2X
i;a
Zia;I þYia;I
hijrjai ð63ị
h0jrjIiRPAẳ ffiffiffi 2 p X
i;a
Zia;I Yia;I
hijrjai ð64ị
Semiempirical SCF MO Methods, Electronic Spectra, and Configurational Interaction 481
An attractive aspect of CIS and RPA is that they both use a common set of MOs for the ground and excited states, which helps in developing qualitative interpretations of the excitation process and in calculation of transition moments. It is straightforward to evaluate hi|r|ai and hi|r|aiprovidediandaboth belong to the same orthonormal MO set. When different MO sets are used for different electronic states (to get the best possible solution), the resultant nonortho- gonality ofifrom one MO set and afrom another MO set causes complications.
RPA can be used with semiempirical methods to calculate electronic spectra.2,69,72 More sophisticated direct methods employ better reference functions (CC, MCSCF, etc.) and/or include more terms in the excitation operatorQI.2
2.38.4.4 Time-dependent Density Functional Theory
Another important method to calculate transition energies and intensities is time-dependent density functional theory (TDDFT) (see Chapter 2.40).
2.38.4.5 Delta Methods 2.38.4.5.1 SCF
When it is possible to describe an excited state by a single configuration wave function, the attractive approach to estimate transition energies is via separate evaluation of SCF energies for both the ground and excited states.2,3,72 Subtraction of the resulting energies gives transition energies in a SCF treatment:
!IẳEIE0 ð65ị
However, in addition to the need to run multiple SCF calculations to obtain the electronic energies, EI, of many excited states, the SCF approach has a number of undesirable features that severely limit its applicability. First, one encounters the problem of variational collapse.2If a molecule possesses no symmetry elements, it is impossible to obtain SCF solutions for any state other than the lowest energy state of a given multiplicity. Second, many excited states cannot be adequately approximated by any single determinant and one must resort to low-spin restricted open-shell approaches which can be difficult to converge.
2.38.4.5.2 Correlated methods
Conceptually one of the simplest ways to study excited states at the correlated level of theory is by means ofEmethods, in which the energy of the system,E, is obtained with some post-HF method.2,3,72While the problem of variational collapse severely complicates efforts to calculate transition energies by SCF, it is not so difficult to obtain CI transition energies, provided the same reference function is used for both states. In this case, the orbitals are usually optimized for the ground state. Correlated energies are then obtained by diagonalization of the Hamiltonian matrix in the basis of the reference determinant and some suitable set of excited configurations. In this way, wave functions for the ground and excited states as well as transition energies are easily obtained. Despite the conceptual simplicity of this approach, it is not often used in practice. All the CI wave functions are based on a common reference state,0, and the MOs making up0 are optimized for the ground state. Since computational considera- tions frequently limitCI approaches toCISD, insufficient orbital relaxation is included, and therefore excited states are not described as well as the ground state. As a result of this imbalance, CISD typically overestimates transition energies. In fact, CIS and RPA are preferable to CISD because these two approaches offer a balanced description of the ground and excited states.2,3
One can avoid the ‘‘orbital problem’’ associated with CI calculations by using MCSCF. The MOs are optimized for each state of interest and the ground and excited states are treated in an even-handed way. Dynamic correlation can then be included by performing MRCI calculations.
This approach can be used to obtain very accurate transition energies. However, it is inconvenient
for the study of multiple excited states because separate orbital optimizations and CI calculations are required for each electronic state. Moreover, the use of different MOs for different electronic states complicates the calculations of transition intensities.
A way to include many of the advantageous features of both the CI and MCSCF-CI approaches is to use ‘‘state-averaged’’ orbitals. In these calculations, a common set of orbitals designed to describe a set of electronic states is used in a subsequent CI calculation. Such a procedure is relatively efficient and is expected to work well, provided the optimal MOs for all the electronic states of interest are not very different. If there is a significant difference between the electronic states, such a state-averaged scheme cannot work too well because all electronic states will be described poorly. In addition, the size of the configuration space required in these calculations is larger than that needed for state-specific MCSCF calculations, resulting in great computational demands.
2.38.4.6 Transition State Method of Slater
This approach to obtain approximate transition energies was proposed by Slater.303,304 It is based on the idea that the total electronic energy of a molecular system in DFT is a function of MO occupation numbers. Instead of basing the calculation on the ground state, the calculation is done for what is called ‘‘the transition state’’: a state in which the occupation numbers for relevant MOs are half-way between those of the initial (ground) and final (excited) state. For example, the difference of the orbital energies, "a"i, calculated for the configuration (. . .) (i )1(i )0.5(a )0.5, which is halfway between the initial electronic state (. . .)(i )1(i )1 and the final state (. . .)(i )1(a )1, is used to estimate the energy of the electronic transition i!a. With the development of other methods (in particular, TDDFT), this approximation is rarely used nowadays.