As methods for ab initio calculations of excited states are becoming increasingly more routine, questions arise concerning how best to extract chemical meaning from such calculations. There are several approaches for analyzing molecular excited states; they are based on reduced one-particle density matrices (OPDMs). The two objects exploited in this analysis are: (i) the difference between the ground- and excited-state OPDMs and (ii) the transition OPDM connecting the ground and excited state. In the case of CIS and TDDFT/TDA wave functions, both quantities are identical and can be directly mapped into the CIS amplitudes; however, for correlated wave functions the two objects are not the same. The most basic analysis includes calculation of attachment/detachment density  and natural transition orbitals . These quantities allow one to arrive to a most compact descrtiption of an excited state. More detailed analysis allows one to derive additional insight about the nature of the excited state. Detailed description and illustrative examples can be found in Refs. [450, 451].
This section describes the theoretical background behind attachment/detachment analysis and natural transition orbitals, while details of the input for creating data suitable for plotting these quantities is described separately in Chapter 10, which also describes additional excited-state analysis tools. For historical reasons, there are duplicate implementations of some features. For example, CIS and TDDFT wave functions can be analyzed using an original built-in code and by using a more recent module, libwfa.
Consider the one-particle density matrices of the initial and final states of interest, and respectively. Assuming that each state is represented in a finite basis of spin-orbitals, such as the molecular orbital basis, and each state is at the same geometry. Subtracting these matrices yields the difference density
Now, the eigenvectors of the one-particle density matrix describing a single state are termed the natural orbitals, and provide the best orbital description that is possible for the state, in that a CI expansion using the natural orbitals as the single-particle basis is the most compact. The basis of the attachment/detachment analysis is to consider what could be termed natural orbitals of the electronic transition and their occupation numbers (associated eigenvalues). These are defined as the eigenvectors defined by
The sum of the occupation numbers of these orbitals is then
where is the net gain or loss of electrons in the transition. The net gain in an electronic transition which does not involve ionization or electron attachment will obviously be zero.
The detachment density
is defined as the sum of all natural orbitals of the difference density with negative occupation numbers, weighted by the absolute value of their occupations where is a diagonal matrix with elements
The detachment density corresponds to the electron density associated with single particle levels vacated in an electronic transition or hole density.
The attachment density
is defined as the sum of all natural orbitals of the difference density with positive occupation numbers where is a diagonal matrix with elements
The attachment density corresponds to the electron density associated with the single particle levels occupied in the transition or particle density. The difference between the attachment and detachment densities yields the original difference density matrix
In certain situations, even the attachment/detachment densities may be difficult to analyze. An important class of examples are systems with multiple chromophores, which may support exciton states consisting of linear combinations of localized excitations. For such states, both the attachment and the detachment density are highly delocalized and occupy basically the same region of space . Lack of phase information makes the attachment/detachment densities difficult to analyze, while strong mixing of the canonical MOs means that excitonic states are also difficult to characterize in terms of MOs.
Analysis of these and other excited states is greatly simplified by constructing Natural Transition Orbitals (NTOs) for the excited states. (The basic idea behind NTOs is rather old , and has been rediscovered several times [449, 453]; the term “natural transition orbitals” was coined in Ref. Martin:2003.) Let denote the transition density matrix from an excited-state calculation. The dimension of this matrix is , where and denote the number of occupied and virtual MOs, respectively. The NTOs are defined by transformations and obtained by singular value decomposition (SVD) of the matrix , i.e. 
The matrices and are unitary and is diagonal, with the latter containing at most non-zero elements. The matrix is a unitary transformation from the canonical occupied MOs to a set of NTOs that together represent the “hole” orbital that is left by the excited electron, while transforms the canonical virtual MOs into a set of NTOs representing the excited electron. (Equivalently, the “holes” are the eigenvectors of the matrix and the particles are eigenvectors of the matrix .) These “hole” and “particle” NTOs come in pairs, and their relative importance in describing the excitation is governed by the diagonal elements of , which are excitation amplitudes in the NTO basis. By virtue of the SVD in Eq. eq:NTO_SVD, any excited state may be represented using at most excitation amplitudes and corresponding hole/particle NTO pairs. [The discussion here assumes that , which is typically the case except possibly in minimal basis sets. Although it is possible to use the transpose of Eq. eq:NTO_SVD to obtain NTOs when , this has not been implemented in Q-Chem due to its limited domain of applicability.]
The SVD generalizes the concept of matrix diagonalization to the case of rectangular matrices, and therefore reduces as much as possible the number of non-zero outer products needed for an exact representation of . In this sense, the NTOs represent the best possible particle/hole picture of an excited state. The detachment density is recovered as the sum of the squares of the “hole” NTOs, while the attachment density is precisely the sum of the squares of the “particle” NTOs. Unlike the attachment/detachment densities, however, NTOs preserve phase information, which can be very helpful in characterizing the diabatic character (e.g., or ) of excited states in complex systems. Even when there is more than one significant NTO amplitude, as in systems of electronically-coupled chromophores , the NTOs still represent a significant compression of information, as compared to the canonical MO basis.
NTOs are available within Q-Chem for CIS, RPA, and TDDFT excited states. For the correlated wave functions (EOM-CC and ADC), they can be computed using libwfa module. The simplest way to visualize the NTOs is to generate them in a format suitable for viewing with the freely-available MolDen or MacMolPlt programs, as described in Chapter 10.