We present a simple quasi-diabatization scheme applicable to spectroscopic studies that can be applied using any wavefunction for which one-electron properties and transition properties can be calculated. The method is based on rotation of a pair (or set) of adiabatic states to minimize the difference between the given transition property at a reference geometry of high symmetry (where the quasi-diabatic states and adiabatic states coincide) and points of lower symmetry where quasi-diabatic quantities are desired. Compared to other quasi-diabatization techniques, the method requires no special coding, facilitates direct comparison between quasi-diabatic quantities calculated using different types of wavefunctions, and is free of any selection of configurations in the definition of the quasi-diabatic states. On the other hand, the method appears to be sensitive to multi-state issues, unlike recent methods we have developed that use a configurational definition of quasi-diabatic states. Results are presented and compared with two other recently developed quasi-diabatization techniques.

## I. INTRODUCTION

The use of quasi-diabatic states and potentials to study chemical phenomena has a long history in chemical physics.^{1–5} Despite the facts that there are innumerable possible quasi-diabatic states (as compared to the unique set of adiabatic states) and that almost all quantum chemistry methods are based on the adiabatic approximation, quasi-diabatic states are ubiquitous in theoretical treatments of electron and energy transfer,^{6–38} scattering and reactive processes,^{39–50} and many spectroscopic problems.^{51–63}

The rigorous use of the term “diabatic” implies that the off-diagonal matrix element of the kinetic energy operator is zero for a pair of states over a range of coordinates. Mead and Truhlar^{3} have shown that such states cannot be globally defined for polyatomic molecules except for the trivial case of geometry-independent basis states. Hence, a more accurate terminology for the type of states used to characterize reactive processes or spectroscopic systems is “quasi-diabatic,”^{53} signifying a set of states where the dominant vibronic coupling occurs in the electronic Hamiltonian and for which the off-diagonal nuclear derivative couplings are small enough to be neglected. In this sense, the intuitive picture of a set of electronic states that preserve their character over a range of important nuclear configurations is a useful guide to the construction of a set of quasi-diabatic states.

The *utility* of a set of quasi-diabatic states is largely determined by how compact the description of the process of interest is when these states are used as a basis. A quasi-diabatic basis would be considered “good” if it provides a description of the process that is accurate and conceptually simpler than in the adiabatic basis or some other quasi-diabatic basis. In a few cases, comparisons of various methods for quasi-diabatization have been made^{64–68} and the similarity of coupling constants from distinct quasi-diabatization schemes provides some evidence for their general utility. In addition, in some cases, comparisons have been made between coupling constants obtained from quasi-diabatization schemes based on physical properties and constants obtained directly from non-adiabatic coupling vectors. The good agreement in these cases further argues for the utility of these schemes for quasi-diabatization.^{67,69}

We have recently introduced a Block Diagonalization (BD) method^{68} (based on the work of Cederbaum, Domcke, and co-workers^{51–53}) for quasi-diabatization of Equation of Motion Coupled Cluster (EOM-CC) states that can treat any number of states, yields quasi-diabatic state couplings and quasi-diabatic potential energy surfaces as a function of geometry, and allows use of general EOM-CC wavefunctions (that is, including triple and higher excitations). Making reference to Eq. (1) (with letters denoting diabatic quantities)

the BD method provides diabatic (*d*) energies (e.g., $EAd+\u2211i\delta Ai\Delta qi+\u2211i\kappa Ai\Delta qi2/2\u2026$) and couplings ($\u2211i\lambda ABi\Delta qi+\cdots $) which are functions of nuclear geometry. Note that if *q* is a non-symmetric coordinate, $\delta iA$ and $\delta iB$ can be chosen to be zero unless the system is subject to the Jahn-Teller effect. Usually, quasi-diabatization methods are concerned with the calculation of *λ _{AB}* (the “coupling constant”) but high-accuracy dynamical treatments may also require the diagonal Hamiltonian elements and higher order corrections to the off-diagonal terms.

^{60,61,63}

The BD method was shown to be (non-trivially) equivalent to a second quasi-diabatization method (denoted QD, for “quasi-diabatic”) developed by one of the authors^{59} in cases where both the BD and QD methods could be applied. One advantage of the BD method is its ability to calculate coupling elements and potential energy surfaces for quasi-diabatic states based on EOM-coupled-cluster singles doubles and triples (CCSDT) wavefunctions where program limitations are such that the QD method cannot be applied. However, one disadvantage of both the QD and BD methods (and any other methods that use configuration expansions to define quasi-diabatic states^{39,51,52,57,66,68}) is the demand that the configurations in which the equation of motion vectors is expanded are equivalent at all relevant geometries. In order to (at least approximately) satisfy this requirement, some form of orbital rotation is required at non-reference geometries.^{52,66,68} Many of the electronic structure methods to which configurational quasi-diabatization approaches have been applied (MCSCF, CI, and EOM-CC) are invariant to orbital rotations among selected classes of orbitals. Thus, it appears that these quasi-diabatization methods depend on making a specific choice for orbital relaxation that the underlying methods do not require. Finally, quasi-diabatization techniques based on configurational definitions of quasi-diabatic states often require gradient calculations^{57,59} or development of orbital and configuration projection codes^{51,52,66,68} that are not readily generalized to other wavefunction types.

With these issues in mind, we present a new method here to calculate quasi-diabatic energies, coupling constants, and properties for spectroscopic problems. The inputs are energies and charge moments, quantities that are routinely calculated in the course of adiabatic quantum chemical calculations. In the examples below, we focus on dipole moments and transition dipole moments, but the method could be applied using any one-electron property that possesses off-diagonal matrix elements for the pair (or set) of states of interest near the geometries of interest. Because the method is based on physical properties, the results are invariant to orbital rotations in the same way that the underlying wavefunctions and energies are. Furthermore, any quantum chemical method that produces charge moments between adiabatic states can be used as input to this quasi-diabatization approach with no special coding, and multiple states can be treated in a single calculation. Finally, the simplicity of the method allows it to be applied with relative ease using a range of quantum chemical approaches providing direct comparisons of coupling constants from a variety of wavefunctions.

The results below suggest that the method is a useful, simple alternative to configuration-based quasi-diabatization schemes, but it is not without limitations. For example, since the method uses transition properties as signatures of state mixing, it can produce qualitatively different quasi-diabatic states compared to configuration-based methods in cases where several quasi-diabatic states are mixed strongly as the geometry departs from a high symmetry point. We discuss how to recognize this situation and calculate quasi-diabatic properties when it occurs. Nevertheless, we believe the simple implementation of the method, its intuitive definition of quasi-diabatic states, and its ability to calculate quasi-diabatic potential energy surfaces and coupling constants at a range of geometries will make it a useful tool for studies of spectroscopic systems that require a multistate interpretation.

## II. MOTIVATION AND METHOD

The method proposed here uses transition moments (TMs) of adiabatic states to determine quasi-diabatic states and we refer to it below as the TM method. Other methods have used transition moments for purposes of quasi-diabatization^{48,65,66,70,71} but in all of these cases the goal was to create quasi-diabatic states with charge transfer, or charge localized, character. The goal here is to use transition moments to create charge *delocalized* quasi-diabatic states, states that are appropriate for many spectroscopic situations as discussed below. We introduce the method using a relatively simple example and then discuss its implementation in more detail.

Consider a molecule with C_{S} symmetry. We are interested in describing the electronic spectrum of this molecule in regions where states of both *A*′ and *A*″ symmetry interact. If there are significant non-adiabatic effects, a multi-state treatment is required to treat the spectrum accurately. In a quasi-diabatic representation, the analogue of Eq. (1) for this system becomes

The sums in the diagonal elements (*i*) run over all normal modes of the system, while the sums in the off-diagonal elements (*j*) only run over *a*″ normal modes, because the two electronic states belong to different irreducible representations of the equilibrium point group of the molecule. In this quasi-diabatic representation, we have assumed that adiabatic and quasidiabatic states coincide at the reference geometry (all Δ*q _{i}* = 0). This is a natural representation for many spectroscopic problems where the ground state equilibrium geometry possesses symmetry and the form of the spectrum is dominated by dynamics in the “Franck Condon” region or nearby geometries.

The dipole moment matrix in C_{S} for this quasi-diabatic representation will have two distinct contributions, one from the components that transform according to the totally symmetric irreducible representation (assumed to be x and y here, with the form of *μ ^{x}* shown below—

*μ*assumes an identical form),

^{y} where the sum over *k* is over totally symmetric modes and that over *j* is, again, over *a*″ modes. A related expression arises for the *A*″ component of the dipole moment operator (assumed to be z here)

Note that because the two electronic states possess different symmetries, the dipole matrices in Eqs. (3) and (4) have different structures. The diagonal elements of the x and y dipole matrices will be non-zero at the reference geometry, while its off-diagonal elements are zero at the reference geometry and small nearby. Conversely, the off-diagonal elements of the z dipole matrix will be large at the reference geometry while its diagonal elements will be zero there, and small near it.

The quasi-diabatic energies and couplings at geometries distinct from the reference geometry are obtained by choosing non-zero values for some or all of the normal coordinates in Eq. (2). The *adiabatic* energies are obtained when the *quasi-diabatic Hamiltonian* (Eq. (2)) is diagonalized at the given geometry. For simplicity, assume that only a single *a*″ mode is displaced and we retain only the leading term in each matrix element. The adiabatic energies at this geometry are (numbers indicate adiabatic quantities while the designations *A*′ and *A*″ refer to diabatic quantities)

Writing the coefficients of the adiabatic states in terms of the quasi-diabatic basis states as

and transforming the z component of the dipole moment matrix into the adiabatic representation, one has (defining $\xi A\u2032j=\u2202\mu A\u2032z/\u2202qj\Delta qj$ and similarly, $\xi A\u2033j=\u2202\mu A\u2033z/\u2202qj\Delta qj$)

The quantities $\xi A\u2032j$ and $\xi A\u2033j$ are variations of the quasi-diabatic dipoles with respect to a normal mode motion perpendicular to the plane of symmetry. Since the quasi-diabatic states have no dipole component in this direction at equilibrium but undergo the same deformation, it is plausible to (roughly) assume that the $\xi A\u2032j$ and $\xi A\u2033j$ have similar sizes. Assuming equality ($\xi A\u2032j=\xi A\u2033j\u2261\xi j$) and utilizing a pair of trigonometric identities, Eq. (6) reduces to

Thus, in the adiabatic representation at the stepped geometry, the off-diagonal elements of the *μ ^{z}* matrix tend to decrease while the difference between the diagonal elements increases.

In applications of conventional quantum chemistry methods, we do not have access to the underlying quasi-diabatic states; rather we calculate adiabatic quantities and attempt to extract quasi-diabatic quantities from them. We propose that quasi-diabatic states can be obtained from the *inverse* of the quasi-diabatic to adiabatic transformation outlined above. If this is the case, then the prescription we advocate to return to quasi-diabatic states is to maximize the off-diagonal dipole matrix elements of the dipole moment component that transforms according to the same irreducible representation as the normal mode of interest. If a mode does not belong to the same irreducible representation as x, y, or z, one can substitute an appropriate quadrupole (or higher) moment for the dipole term and perform a similar transformation.

Thus, in the TM method, one selects a non-totally symmetric normal coordinate that couples the two states of interest and steps the geometry away from the equilibrium (higher-symmetry) geometry along this mode by an amount Δ*q*. At this new (lower-symmetry) geometry, the energies and the components of the dipole moments and transition dipole moments that belong to the same irreducible representation as the normal coordinate of interest are calculated. At this new geometry, the relevant quantities are

With reference to the model introduced above, we emphasize that at the reference geometry the adiabatic dipoles would be zero in this direction and the off-diagonal elements would have larger magnitudes than at the stepped geometry. To transform to the quasi-diabatic states at the stepped geometry, we now seek a rotation in the two state space to maximize the transition dipole moment,

We denote the transformation angle *θ* to distinguish it from the angle *ϕ* that transforms from quasi-diabatic to adiabatic states (see above) since these angles can be different when $\xi A\u2032j\u2260\xi A\u2033j$. Thus, in terms of *θ*, the transformed dipole matrix has as its off-diagonal element,

Taking the derivative with respect to *θ* and setting the result equal to zero yields

Rearranging the above one obtains

This value of *θ _{max}* leads to the maximum value of the absolute value of $\mu ABd$ at the stepped geometry. Applying this transformation to the energy matrix, one obtains

which can be written using another trigonometric relation as

Equation (16) is the central result for the method developed here. Note that in Eq. (16), all components on the right-hand side are *adiabatic* quantities that are naturally obtained from general quantum chemistry codes. Finally, division by the normal mode step size (Δ*q*) in the limit of small step size yields

The TM method described above can be applied using the output of any quantum chemistry code that provides energies and (diagonal and off diagonal) charge moments for a set of states. Note that the TM method’s definition of the quasi-diabatic states is the pair or set of states that possess maximal transition moments (for the component of the transition and dipole moments belonging to irreducible representation of the coupling mode) and that this definition can be applied at any step along the normal coordinate. As a result, the method allows step-wise construction of quasi-diabatic potentials.

We have introduced the TM method using as an example a two-state system but it can be applied to multi-state problems as well. Below we treat NO_{2} using up to four adiabatic states. In this multi-state example, we again choose a single normal mode, stepped along the coordinate by an amount Δ*q*, and use the TM prescription to quasi-diabatize the four states via successive pairwise Jacobi rotations. In this case, two states are of ^{2}A_{1} symmetry and two of ^{2}B_{2} symmetry. The Jacobi rotations maximize the transition dipole matrix elements between states correlating with adiabatic states from different irreducible representations of the equilibrium geometry and set to zero the transition dipoles for states correlating with adiabatic states from the same irreducible representation. This procedure converges quickly in this case and can be generalized to any number of states for which adiabatic quantities can be obtained.

A more subtle application of quasi-diabatization using the TM approach is found in certain Jahn-Teller systems. An example of this kind is found in the NO_{3} radical, where one may be interested in the interaction between the Jahn-Teller degenerate states and another non-degenerate state. Not surprisingly, we found that we need to treat both components of the dipole operator that belong to the irreducible representation of the degenerate normal mode. To apply the TM method, we use the following approach:

We calculate the dipole and transition moments for both Cartesian coordinates belonging to the irreducible representation of the normal mode of interest.

In our calculation, one of the two Cartesian directions is

*parallel*to the direction of the motion associated with the normal coordinate of interest. Before applying the TM method, we diagonalized the 2 × 2 dipole moment matrix for the Jahn-Teller states for the component of the dipole moment of the degenerate irreducible representation*perpendicular*to the normal coordinate of interest. This transformation is then applied to the energy matrix and the adiabatic dipole moment matrix for the parallel coordinate. This ensured that the coupling between JT states was now strictly in the parallel component of the dipole moment for the TM method.A series of pairwise Jacobi rotations amongst the three states of interest then led to quasi-diabatic states for the JT problem.

One challenge that the TM method faces is that it uses a single property to calculate quasi-diabatic states. In our case, the “signals” of adiabatic mixing with geometry are changes in dipole moments and transition moments. However, if several states interact, it may be that the change in one adiabatic state’s diagonal dipole moment could arise from interactions with several states, not merely the other state one has chosen for a two-state treatment. In Eq. (6), this would mean that contributions from interactions with several states would contribute to the dipole moment (diagonal elements) at a non-symmetric geometry for a given adiabatic state. One might not anticipate this situation, and as a result select only a pair of states for quasi-diabatization. However, using the model of Eqs. (6) and (8), one can arrive at a signature of these multi-state effects. If a two-state model is appropriate, then the diagonal elements of the dipole matrix in the quasi-diabatic basis at the displaced geometries should be considerably smaller than the corresponding adiabatic quantities (as indicated by comparing the dipole matrices in Eqs. (4) and (8)). On the other hand, if the magnitudes of the dipole moment of the quasi-diabatic and adiabatic states from which they are formed are similar in size, it means that the quasi-diabatic states most similar to the high-symmetry adiabatic states are most likely composed of more than the pair of adiabatic states selected in the TM treatment. Thus, the extent to which the diagonal dipole moments are reduced can be used as an indicator of multi-state effects. We illustrate the use of this diagnostic for NO_{2} in Section IV.

In the TM approach, we focus on the dipole matrix component that belongs to the same irreducible representation as the non-totally symmetric normal coordinate of interest. The above model naturally shows that it is this dipole component that exhibits the greatest change with respect to motion along the normal coordinate. One might question whether the other Cartesian dipole components would have an impact on the quasi-diabatic state definition if they were included in the treatment for quasi-diabatization. In addition, for systems where quasi-diabatic states are sought at points of low or no symmetry, the TM approach is not immediately applicable, since it is not clear that the transition moment should be maximized along coordinates that show asymmetric variation in one-electron properties. To treat cases of these types, we have developed the method presented in the Appendix. It can be viewed as an Inverse Boys (IB) transformation^{64,72,73} when applied to spectroscopic problems (Boys “delocalization,” rather than Boys localization) and is based on minimization of the summed squared dipole moment differences between states at the stepped and reference geometries. In addition to describing the method, we present results illustrating its use. The results of the IB approach support the TM method’s use of a single dipole component when symmetry is present. The IB method also provides a more general approach applicable to situations when no symmetry is present.

The reader may have noted a similarity between the TM method and the Werner and Meyer^{70} (WM) quasi-diabatization method or the Generalized Mulliken-Hush (GMH) method^{71} used in electron transfer problems. Unlike either of these methods, which set the transition dipole moment connecting quasi-diabatic states to zero, the TM method *maximizes* the transition dipole moment connecting quasi-diabatic states. A naive perspective would assume that it is, therefore, impossible for the WM/GMH and TM methods to both yield quasi-diabatic states but this is to miss the point of quasi-diabatic states and their utility. Werner and Meyer sought charge localized states since they were interested in a charge transfer problem (the same construction is used in the GMH approach). The H and dipole matrices analogous to those of Eqs. (1) and (3) for the WM/GMH quasi-diabatic states are

and

where K and L denote a pair of charge localized states that belong to the *same* irreducible representation and *z* is the dipole moment component belonging to this irreducible representation. The assumption of zero off-diagonal component connecting K and L stems from the charge localized nature of the quasi-diabatic states and the weak distance-dependence of the dipole moment operator.^{66,71} The WM/GMH approach thus transforms from adiabatic states to quasi-diabatic states by diagonalizing the (parallel) adiabatic dipole moment matrix, yielding states that are as charge-localized as possible within the n-state space of interest. In a two state space, this transformation is equivalent to maximizing the dipole moment difference between the pair of quasi-diabatic states. On the other hand, in the TM method, we seek states that are as *delocalized* as possible, since they correlate with the adiabatic states at points of high symmetry. The TM quasi-diabatic states have as small a dipole moment difference as possible, corresponding to their delocalized nature. Thus, the TM approach is essentially the inverse of the WM/GMH approaches.

## III. COMPUTATIONAL METHODS

All EOM-CC calculations were performed with the CFOUR suite of electronic structure theory codes.^{74} Configuration interaction calculations on BNB were performed with GAMESS.^{75}

The EOM-CC calculations for BNB, NO_{2}, NO_{3}, and HCO_{2} were done as EOMIP CC calculations. The SCF orbitals were those of the anion in each case. The basis sets used were BNB, NO_{2} and HCO_{2}: DZP,^{76} and NO_{3}: ANO0.^{63,77} All reference geometries were calculated using CCSD analytic gradients on the parent anion. Parent anion normal modes/vibrational frequencies were calculated by CCSD analytic second derivatives. The CI calculations used the CCSD geometries and were based on SDT-CI or SDTQ-CI results for the neutral doublet states expressed in the RHF-SCF orbitals of the ground state anion.

Based on the dimensionless normal modes for the parent geometry, steps were taken along relevant normal coordinates for the calculation of the coupling constant. A similar approach was used in our development of the BD method and details can be obtained there.^{68} We compare results from the TM method with those obtained from the BD and QD approaches below.^{59}

## IV. RESULTS

We illustrate the application of the TM method to the ^{2}Σ_{g}^{+} and ^{2}Σ_{u}^{+} states of BNB using the results in Table I. As described above inputs to the TM method are the adiabatic energies and dipole matrix elements for the states of interest at the stepped geometry. We treat coupling induced by motion along the antisymmetric σ_{u} coordinate and *z* is the dipole component that belongs to the σ_{u} irreducible representation in D_{∞h}. While the TM method only requires adiabatic quantities at the stepped geometry, we include data from the reference geometry to illustrate the changes induced by the geometry step. At the stepped geometry, the two adiabatic states have non-zero dipole moments with opposite signs, signaling polarization in two distinct directions. Note also that given the linear geometry of BNB and the vibrational coordinate we chose, the only nonzero components of the dipole and transition moments occur along the internuclear axis (z).

Quantity . | Geometry . | ^{2}Σ_{g}^{+} value
. | ^{2}Σ_{u}^{+} value
. |
---|---|---|---|

μ (z) | Equilibrium | 0.000 0 | 0.000 0 |

μ_{12} (z) (non-GM) | Equilibrium | 2.069 8/2.000 4 | |

Energy | Equilibrium | −104.016 08 | −104.043 53 |

μ (z) | 0.1 σ_{u} | 0.203 9 | −0.156 0 |

μ_{12} (z) (non-GM) | 0.1 σ_{u} | −2.061 8/-1.992 4 | |

μ_{12} (z) GM | 0.1 σ_{u} | 2.026 8 | |

Energy | 0.1 σ_{u} | −104.015 99 | −104.043 54 |

Quasi-diabatic μ (z) | 0.1 σ_{u} | 0.024 0 | 0.024 0 |

Quasi-diabatic μ (z) _{AB} | 0.1 σ_{u} | 2.034 8 | |

Quasi-diabatic energy | 0.1 σ_{u} | −104.016 04 | −104.043 49 |

H _{AB} | 0.1 σ_{u} | 0.001 22 | |

λ (cm_{AB}^{−1}) | 0.1 σ_{u} | −2673.7 |

Quantity . | Geometry . | ^{2}Σ_{g}^{+} value
. | ^{2}Σ_{u}^{+} value
. |
---|---|---|---|

μ (z) | Equilibrium | 0.000 0 | 0.000 0 |

μ_{12} (z) (non-GM) | Equilibrium | 2.069 8/2.000 4 | |

Energy | Equilibrium | −104.016 08 | −104.043 53 |

μ (z) | 0.1 σ_{u} | 0.203 9 | −0.156 0 |

μ_{12} (z) (non-GM) | 0.1 σ_{u} | −2.061 8/-1.992 4 | |

μ_{12} (z) GM | 0.1 σ_{u} | 2.026 8 | |

Energy | 0.1 σ_{u} | −104.015 99 | −104.043 54 |

Quasi-diabatic μ (z) | 0.1 σ_{u} | 0.024 0 | 0.024 0 |

Quasi-diabatic μ (z) _{AB} | 0.1 σ_{u} | 2.034 8 | |

Quasi-diabatic energy | 0.1 σ_{u} | −104.016 04 | −104.043 49 |

H _{AB} | 0.1 σ_{u} | 0.001 22 | |

λ (cm_{AB}^{−1}) | 0.1 σ_{u} | −2673.7 |

Due to the non-Hermitian nature of the EOM-CC approach, the 1-2 and 2-1 transition dipole moments are not equal. We use the geometric mean (GM) of the transition moments as input to the TM method. When the transformation that maximizes the transition moment (Eqs. (11) and (14)) is applied to the adiabatic dipole moment matrix, the resulting quasi-diabatic transition moment increases relative to the adiabatic transition moment and the quasi-diabatic (diagonal) dipole moments are significantly smaller than the corresponding adiabatic dipole moments. This behavior nicely illustrates the differences between quasi-diabatic and adiabatic states discussed in Eqs. (1)–(8).

Based on the quantities in Table I, Table II presents a comparison of TM results using various wavefunctions and compares TM results with analogous results from the QD and BD approaches. We use identical EOM-CC methodologies for the BD, QD, and TM EOM results and also include CI results using the TM method. All of the coupling constants are within 7% of each other and the TM, BD, and QD results based on EOM-IP-CCSD wavefunctions are within 0.4% of each other. The small difference between the QD and BD results is due to the finite step size chosen in the BD calculation. In the limit of zero BD step size, the BD and QD methods would yield identical coupling constants.

Wavefunction . | Diabatization method . | λ (cm_{AB}^{−1})
. |
---|---|---|

EOM-IP-CCSD | TM | 2674 |

EOM-IP-CCSD | QD | 2684 |

EOM-IP-CCSD | BD | 2683 |

CISDT | TM | 2504 |

CISDTQ | TM | 2535 |

Wavefunction . | Diabatization method . | λ (cm_{AB}^{−1})
. |
---|---|---|

EOM-IP-CCSD | TM | 2674 |

EOM-IP-CCSD | QD | 2684 |

EOM-IP-CCSD | BD | 2683 |

CISDT | TM | 2504 |

CISDTQ | TM | 2535 |

Table III presents comparisons of results based on the TM, BD, and QD methods for two states of the HCO_{2} radical along two different normal coordinates. In this example, the molecule lies in the yz plane and z is the C_{2} axis. We consider motion along the b_{2} normal coordinates, which belong to the same irreducible representation as y in C_{2V}. The results of Table III show that agreement is excellent amongst the results from the various methods. Note that a step size of “0” means that we have extrapolated the finite step size results to zero step size using a polynomial fit. Because the BD and TM approaches use stepped geometries in order to calculate coupling elements, they also yield quasi-diabatic state energies at geometries other than the equilibrium geometry. We have used these quasi-diabatic energies to calculate the corresponding quasi-diabatic vibrational frequencies for the TM potentials and compare them with the previously obtained BD results.^{68} Modest differences exist between the TM and BD quantities, with the largest occurring for BD 1b_{2} quasi-diabatic harmonic frequencies. After the transition to quasi-diabatic states, the dipole moments are reduced considerably relative to the adiabatic dipole moments at the stepped geometry (0.1 steps, 1b_{2}: adiabatic = 0.178/−0.168; quasi-diabatic: 0.005/0.005; 2b_{2}: adiabatic = − 0.449/0.465; quasi-diabatic: 0.008/0.008). This behavior is similar to that observed for BNB.

Mode . | Method . | Step size . | λ (cm_{AB}^{−1})
. | ω (cm^{−1})
. |
---|---|---|---|---|

1b_{2} | TM | 0.1 | 682.3 | |

TM | 0.3 | 683.1 | ||

TM | 0 | 682.2 | 955/1 499 | |

BD | 0.1 | 667.8 | ||

BD | 0.3 | 668.0 | ||

BD | 0 | 667.8 | 889/1 565 | |

QD | … | 667.8 | ||

421i/2 876^{a} | ||||

2b_{2} | TM | 0.1 | 2192.0 | |

TM | 0.3 | 2188.3 | ||

TM | 0 | 2192.4 | 1 024/1 477 | |

BD | 0.1 | 2198.6 | ||

BD | 0.3 | 2195.2 | ||

BD | 0 | 2199.1 | 1 124/1 377 | |

QD | … | 2199.1 | ||

79 431i/10 444^{a} |

Mode . | Method . | Step size . | λ (cm_{AB}^{−1})
. | ω (cm^{−1})
. |
---|---|---|---|---|

1b_{2} | TM | 0.1 | 682.3 | |

TM | 0.3 | 683.1 | ||

TM | 0 | 682.2 | 955/1 499 | |

BD | 0.1 | 667.8 | ||

BD | 0.3 | 668.0 | ||

BD | 0 | 667.8 | 889/1 565 | |

QD | … | 667.8 | ||

421i/2 876^{a} | ||||

2b_{2} | TM | 0.1 | 2192.0 | |

TM | 0.3 | 2188.3 | ||

TM | 0 | 2192.4 | 1 024/1 477 | |

BD | 0.1 | 2198.6 | ||

BD | 0.3 | 2195.2 | ||

BD | 0 | 2199.1 | 1 124/1 377 | |

QD | … | 2199.1 | ||

79 431i/10 444^{a} |

^{a}

Adiabatic frequencies based on analytic first derivatives.

The results in Tables IV and V for NO_{2} provide a counter-example to the “simple” two-state mixing observed above. Once again the molecule is taken to lie in the yz plane and the symmetry axis is z. The data in Table IV show that the y and z moments both change at the stepped geometry (relative to their reference values) but the vast majority of this change occurs in the y components and the TM method utilizes only the dipole components in the y direction.

Quantity . | Geometry . | ^{2}A_{1} value
. | ^{2}B_{2} value
. |
---|---|---|---|

μ (y)/μ (z) | Equilibrium | 0/0.180 0 | 0/0.023 4 |

μ_{12} (y)/μ_{12} (z) (non-GM) | Equilibrium | 0.510 5/0 0.488 1/0 | |

Energy | Equilibrium | −204.620 02 | −204.585 00 |

μ (y)/μ (z) | 0.1 1b_{2} | 0.006 7/0.179 9 | 0.011 9/0.023 6 |

μ_{12} (y)/μ_{12} (z) (non-GM) | 0.1 1b_{2} | 0.510 5/0.005 1 | 0.488 1/0.004 8 |

μ_{12} (y) GM | 0.1 1b_{2} | 0.499 2 | |

Energy | 0.1 1b_{2} | −204.620 00 | −204.584 99 |

Quasi-diabatic μ (y) | 0.1 1b_{2} | 0.009 3 | 0.009 3 |

Quasi-diabatic μ (y) _{AB} | 0.1 1b_{2} | 0.499 2 | |

Quasi-diabatic energy | 0.1 1b_{2} | −204.620 00 | −204.584 99 |

H _{AB} | 0.1 1b_{2} | 0.000 090 55 | |

λ (cm_{AB}^{−1}) | 0.1 1b_{2} | −198.7 |

Quantity . | Geometry . | ^{2}A_{1} value
. | ^{2}B_{2} value
. |
---|---|---|---|

μ (y)/μ (z) | Equilibrium | 0/0.180 0 | 0/0.023 4 |

μ_{12} (y)/μ_{12} (z) (non-GM) | Equilibrium | 0.510 5/0 0.488 1/0 | |

Energy | Equilibrium | −204.620 02 | −204.585 00 |

μ (y)/μ (z) | 0.1 1b_{2} | 0.006 7/0.179 9 | 0.011 9/0.023 6 |

μ_{12} (y)/μ_{12} (z) (non-GM) | 0.1 1b_{2} | 0.510 5/0.005 1 | 0.488 1/0.004 8 |

μ_{12} (y) GM | 0.1 1b_{2} | 0.499 2 | |

Energy | 0.1 1b_{2} | −204.620 00 | −204.584 99 |

Quasi-diabatic μ (y) | 0.1 1b_{2} | 0.009 3 | 0.009 3 |

Quasi-diabatic μ (y) _{AB} | 0.1 1b_{2} | 0.499 2 | |

Quasi-diabatic energy | 0.1 1b_{2} | −204.620 00 | −204.584 99 |

H _{AB} | 0.1 1b_{2} | 0.000 090 55 | |

λ (cm_{AB}^{−1}) | 0.1 1b_{2} | −198.7 |

Mode . | Method . | Step size . | λ (cm_{AB}^{−1})
. |
---|---|---|---|

1b_{2} | TM 2-state | 0.1 | 199 |

TM 2-state | 0.05 | 202 | |

TM 4-state | 0.1 | 641/1741/4148/849^{a} | |

BD 2-state | 0.1 | 638 | |

QD-pairwise 2-state | … | 638/3744/4806/196^{a} |

Mode . | Method . | Step size . | λ (cm_{AB}^{−1})
. |
---|---|---|---|

1b_{2} | TM 2-state | 0.1 | 199 |

TM 2-state | 0.05 | 202 | |

TM 4-state | 0.1 | 641/1741/4148/849^{a} | |

BD 2-state | 0.1 | 638 | |

QD-pairwise 2-state | … | 638/3744/4806/196^{a} |

^{a}

Couplings are between quasi-diabatic states 1-2, 1-3, 2-4, and 3-4, with the 1-2 coupling corresponding to the *λ _{AB}* values obtained in the 2-state approaches. In the 2-/4-state cases using the TM method, the final quasi-diabatic

*μ*(y) for all states are 0.0093/0.016, respectively.

In Table V, we compare two-state TM, BD, and QD calculations of the coupling constant between the ^{2}A_{1} and ^{2}B_{2} states of NO_{2}. There are significant differences between the two-state TM results and those of the other methods. These differences do not diminish when a smaller step size is used; halving the step size does very little to ameliorate the disagreement. Clearly, the TM method yields different quasi-diabatic states than the QD or BD method here. Based on the discussion surrounding Eqs. (1)-(8), this discrepancy was foreshadowed by the similar adiabatic and TM quasi-diabatic dipole moments (in the y direction) shown in Table IV for the two-state NO_{2} calculations. This small change in the dipole moment upon quasi-diabatization could stem either from distinctly different dipole moments along the b_{2} normal coordinate for the underlying quasi-diabatic states or mixing with other, higher lying quasi-diabatic states at the stepped geometry. In order to probe this difference between the methods, we performed TM calculations using 4 adiabatic states. The 4-state results are encouraging; they produce a 1^{2}A_{1}-1^{2}B_{2} coupling constant similar to that obtained using the QD and BD results. This calculation uses pair-wise Jacobi rotations (based on Eqs. (11) and (14)) on the full four state energy and projected dipole moment matrices until convergence is reached. We also obtain couplings between the 1^{2}A_{1} and 1^{2}B_{2} states and higher ^{2}A_{1} and ^{2}B_{2} states. It would be interesting to explore further expansion of the adiabatic state space but we were unable to do so in a balanced way for NO_{2} due to convergence issues for higher adiabatic states in the small basis used. At present we can say that the TM method allows relatively simple extension to multi-state treatments when the electronic structure results can be obtained, and that the relative size of the quasi-diabatic and adiabatic dipole moments can be used as a signature of multi-state effects. The fact that NO_{2} is not well-described as a two-state system was first noted by Ichino *et al.*^{59}

Table VI presents results from another multi-state system, namely, the coupling elements between the X ^{2}A_{2}′ and ^{2}E′ states of NO_{3} along the 1e′ and 2e′ normal coordinates defined by the ground state of the NO_{3} anion. Here, however, the results are in better agreement with the BD and QD methods. As discussed above, in this case, we prediagonalize one component of the dipole moment matrix before applying the TM method on the other component of the doubly degenerate irreducible representation. The prediagonalization is necessitated because the two components of the ^{2}E′ states (when resolved into a_{1} and b_{2} in the C_{2V} symmetry subgroup of D_{3h}) are connected to the X state by distinct components of the in-plane dipole operators at the reference geometry. At the stepped geometry, when full symmetry is broken, a large transition dipole element connecting the ^{2}E′ components is induced along the non-stepped direction due to a second-order interaction between the quasi-diabatic states mediated by the X state. Prediagonalization removes this interaction and the standard multi-state TM procedure can then be applied to NO_{3}.

Mode . | Method . | Step size . | λ (eV)
. _{AB} |
---|---|---|---|

1e′ | TM | 0.1 | 0.354 |

BD | 0.1 | 0.380 | |

QD | … | 0.379 | |

2e′ | TM | 0.1 | 0.123 |

BD | 0.1 | 0.140 | |

QD | … | 0.140 |

Mode . | Method . | Step size . | λ (eV)
. _{AB} |
---|---|---|---|

1e′ | TM | 0.1 | 0.354 |

BD | 0.1 | 0.380 | |

QD | … | 0.379 | |

2e′ | TM | 0.1 | 0.123 |

BD | 0.1 | 0.140 | |

QD | … | 0.140 |

The results for NO_{3} are in fairly good agreement with the BD and QD results, differing by no more than 13% for the two modes. The values reported for the TM results are the square root of the sum of the squares of the coupling constants between the X and individual ^{2}E′ components. The simplicity and accuracy of the TM method for a system with considerable complexity is encouraging.

## V. DISCUSSION

We have presented a simple, physically motivated, method for calculating quasi-diabatic coupling elements and potential energy surfaces using adiabatic energies and dipole moments. The TM method uses maximization of the component of the transition moment that belongs to the same irreducible representation as the normal mode coordinate of interest. The method can treat multiple states and the easily obtained results are generally close to those obtained from the completely independent and more involved QD and BD approaches.

The principal advantage of this method is that it is based on quantities that are easily obtained from conventional quantum chemistry codes without extensive programming (unlike the BD and QD approaches). As a result, it can be used to explore quasi-diabatic couplings based on a range of wavefunctions with little overhead. In addition, one obtains quasi-diabatic state energies essentially for free in the TM method (as in the BD method) and thus the behavior of the quasi-diabatic potential energy surfaces is accessible with the TM approach. In most of the cases discussed above, the agreement with the QD or BD approaches is quite encouraging.

On the other hand, the TM method has a number of potential drawbacks when compared with other methods. First, because the method does not use a configurational definition of the quasi-diabatic states, it is unable to “isolate” pairs of states in the quasi-diabatization procedure. That is, since the only parameters determining the resulting quasi-diabatization are the dipole and transition dipole moments, one does not have control over how many quasi-diabatic states contribute, at the stepped geometry, to these quantities. NO_{2} is an excellent example of this behavior and illustrates that without a state projection scheme (*à la* the BD approach), the resulting TM “quasi-diabatic states” may be mixtures of several “pure” quasi-diabatic states (as defined at the reference geometry). The method has a natural way to gauge the importance of this effect (the relative sizes of the adiabatic and quasi-diabatic dipole moments at the stepped geometry), and the remedy is a multi-state treatment. As shown in NO_{3}, an appropriate multi-state treatment can be quite effective.

In addition, the TM method requires calculation of the relevant dipole matrix elements. For higher order EOM-CC treatments (including triple and higher excitations), these property calculations may be significantly more demanding to implement. It is this fundamental difficulty that drove us to develop the BD approach to calculate quasi-diabatic quantities using EOM-CC with triple and higher excitations.

The differences between the TM and BD/QD approaches in the case of NO_{2} (or other similar multi-state systems) raise the question of which methods yield the “correct” quasi-diabatic states. Ideally this would be answered by comparison with states (and coupling constants) obtained by diagonalizing the nonadiabatic coupling element along a given normal coordinate. At present we are unable to exactly calculate the non-adiabatic coupling element for these EOM-CC wavefunctions, so a direct answer cannot be given to this question. However, comparisons of quasi-diabatic quantities obtained from the GMH method have been made with those obtained by diagonalizing the nonadiabatic coupling (the initial conditions of the differential equation determining the geometry-dependent rotation angle define the diabatic states obtained) and excellent agreement has been obtained.^{67,69} While the GMH method (as outlined above) is the inverse of the TM approach, it is, nevertheless, a transformation based on dipole and transition dipole moments to a set of quasi-diabatic states consistent with the zeroth-order states of interest. As a result, it is reasonable to assume that the TM method will be similarly accurate in the two-state limit.

Despite these issues, we believe that the accuracy and ease of implementation of this method will make it a useful tool in many cases where multi-state effects (greater than 2 states) are unimportant or at least not dramatic. While the TM method certainly does not render the QD or BD method obsolete it will be a useful first line of attack (and frequently sufficient for the system of interest) for quasi-diabatization in spectroscopic systems. In addition, because it provides a simple means to compare quasi-diabatic quantities derived from different wavefunctions, it will be quite useful for testing orbital- and excitation-level-effects on quasi-diabatic quantities.

## VI. CONCLUSIONS

We present a new quasi-diabatization method applicable to spectroscopic studies that can be applied using any quantum chemical method for which one-electron properties and transition properties can be calculated. Compared to other quasi-diabatization techniques, it requires no special programming, allows direct comparison between quasi-diabatic quantities calculated using different types of wavefunctions, and is free of any selection of configurations in the definition of the quasi-diabatic states. On the other hand, it appears to be more sensitive to multi-state issues than recent methods we have developed that use a configurational definition of quasi-diabatic states. In most cases, the results are in excellent agreement with those from completely independent quasi-diabatization schemes and we present a diagnostic for determining when multi-state effects are present.

## Acknowledgments

R.J.C. is grateful to Harvey Mudd College for ongoing financial support. J.F.S. gratefully acknowledges support from the National Science Foundation No. CHE-1361031. Additional financial support was provided through the NSF-ROA program; R.J.C. and J.F.S. are pleased to acknowledge this support.

### APPENDIX: INVERSE BOYS METHOD

As noted in the main body of the text, there are often higher-order changes in dipole and transition moment components that do not belong to the irreducible representation of the normal coordinate of interest. While their impacts on the results of the TM method are expected to be negligible, it would be advantageous to have a method that could assess their importance. Alternatively, it may be that quasi-diabatic states are desired at a point of no symmetry, where transition dipole maximization is not an appropriate characteristic with which to define quasi-diabatic states. Despite the lack of symmetry, it is clear that the quasi-diabatic states possess properties similar to those of the various states of interest at the reference geometry. With this in mind, a simple approach satisfying this “similarity” criterion can be developed using essentially the inverse of Boys localization at the state level.^{65}

The essence of the approach for a two-state system is that one minimizes R,

where $\mu \iota \iota d\u20d7$ is the dipole vector of quasi-diabatic state *i* at the stepped geometry and $\mu \iota \iota 0\u20d7$ is the corresponding quantity at the reference geometry. Here, we use numbers to indicate both adiabatic and quasi-diabatic states to emphasize the correlation of the quasi-diabatic states with the adiabatic states at the reference geometry and use the superscript *d* to indicate quasi-diabatic values. The quasi-diabatic states are defined as those that have dipole moments as similar as possible to the dipole moments of the references states defined at the reference geometry. In what follows we also require the components of $\mu \iota \iota \u20d7$, the dipole vector of the adiabatic state *i* at the stepped geometry.

As in the TM method, we use the angle *θ* to parameterize the state rotation (the $\mu \iota \iota d\u20d7$ depend on *θ*). The derivative of Eq. (A1) with respect to *θ* is

which is set equal to zero. In the two state case, one can show that

and Eq. (A2) becomes

Since this is a non-linear optimization, a numerical search for *θ* is required. For more than two states, a series of Jacobi rotations need to be performed in pairwise fashion, as is done in the usual Boys localization procedure.^{64} Equation (A4) is the final working equation for what we term the IB method.

There are two potential advantages of the IB approach when compared with the TM method presented above. First, the inverse Boys approach does not maximize the transition moment in an unconstrained fashion so it can handle situations where the derivative of the given transition property is not zero at the reference geometry. Second, the inverse Boys method does not require use of a single dipole component since it minimizes the squared dipole difference, which includes all dipole components. While the TM method was shown above to yield excellent agreement with other quasi-diabatization schemes, there may be occasions for which the inverse Boys approach is useful and so we present results comparing the two methods below.

In order to illustrate the relationship between the TM and IB methods, we now make two further assumptions with regard to the states and quantities used in Eq. (A1). If one is treating a system with high symmetry and focuses only on the dipole component belonging to the same irreducible representation as the non-totally symmetry normal mode, Eq. (A4) reduces to

Equation (A5) can be satisfied when either

or when

Equation (A6) is identical to condition given in Eq. (13), i.e., the maximized transition moment is equivalent to there being no difference between quasi-diabatic dipole moments (Eq. (A1)). Equation (A7) is the condition that the quasi-diabatic dipole moment difference is maximized, i.e., the transition moment is zero between quasi-diabatic states. This latter condition is equivalent to the quasi-diabatization method of Werner and Meyer and the generalized Mulliken-Hush method. The IB method, therefore, provides a clear connection between both the WM/GMH and TM approaches and a prescription by which each can be derived.

We have applied the inverse Boys method to two of the systems considered above. Tables VII and VIII present results for HCO_{2} and NO_{2} showing that the TM method’s restriction to a single dipole component has essentially no impact on these results. These results also demonstrate that the inverse Boys method can treat all components of the dipole moment and transition dipole moment even when the reference geometry has a nonzero dipole moment (both of the systems considered here have a nonzero dipole along their respective C_{2} axes). This feature of the inverse Boys method may make it particularly useful when defining quasi-diabatic states for systems of low or no symmetry at their reference geometry. We shall make further investigations along this direction in the future.

Mode . | Method . | Step size . | λ (cm_{AB}^{−1})
. |
---|---|---|---|

1b_{2} | TM | 0.1 | 682.3 |

Inverse Boys | 0.1 | 681.5 | |

2b_{2} | TM | 0.1 | 2192.0 |

Inverse Boys | 0.1 | 2192.0 |

Mode . | Method . | Step size . | λ (cm_{AB}^{−1})
. |
---|---|---|---|

1b_{2} | TM | 0.1 | 682.3 |

Inverse Boys | 0.1 | 681.5 | |

2b_{2} | TM | 0.1 | 2192.0 |

Inverse Boys | 0.1 | 2192.0 |

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