# 7.13.2 Natural Transition Orbitals

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.518 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 old593 and has been rediscovered several times;615, 623 these orbitals were later shown to be equivalent to CIS natural orbitals.894) Let $\mathbf{T}$ denote the transition density matrix from an excited-state calculation. The dimension of this matrix is $O\times V$, where $O$ and $V$ denote the number of occupied and virtual MOs, respectively. The NTOs are defined by transformations $\mathbf{U}$ and $\mathbf{V}$ obtained by singular value decomposition (SVD) of the matrix $\mathbf{T}$, i.e.,623

 $\mathbf{UTV}^{\dagger}=\bm{\Lambda}$ (7.109)

The matrices $\mathbf{U}$ and $\mathbf{V}$ are unitary and $\bm{\Lambda}$ is diagonal, with the latter containing at most $O$ non-zero elements. The matrix $\mathbf{U}$ 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 $\mathbf{V}$ transforms the canonical virtual MOs into a set of NTOs representing the excited electron. (Equivalently, the “holes” are the eigenvectors of the $O\times O$ matrix $\mathbf{TT}^{\dagger}$ and the particles are eigenvectors of the $V\times V$ matrix $\mathbf{T}^{\dagger}\mathbf{T}$.615) These “hole” and “particle” NTOs come in pairs, and their relative importance in describing the excitation is governed by the diagonal elements of $\bm{\Lambda}$, which are excitation amplitudes in the NTO basis. By virtue of the SVD in Eq. (7.109), any excited state may be represented using at most $O$ excitation amplitudes and corresponding hole/particle NTO pairs. (The‘ discussion here assumes that $V\geq O$, which is typically the case except possibly in minimal basis sets. Although it is possible to use the transpose of Eq. (7.109) to obtain NTOs when $V, 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 $\mathbf{T}$. 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., $\pi\pi^{\ast}$ or $n\pi^{\ast}$) of excited states in complex systems. Even when there is more than one significant NTO amplitude, as in systems of electronically-coupled chromophores,518 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, TDDFT, ADC, and EOM-CC methods. 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 11.