For electron transfer (ET) and excitation energy transfer (EET) processes, the electronic coupling is one of the important parameters that determine their reaction rates. For ET, Q-Chem provides the coupling values calculated with the generalized Mulliken-Hush (GMH),141 fragment-charge difference (FCD),999 Boys localization,932 and Edmiston-Ruedenbeg928 localization schemes. For EET, options include fragment-excitation difference (FED),403 fragment-spin difference (FSD),1088 occupied-virtual separated Boys localization,931 or Edmiston-Ruedenberg localization.928 In all these schemes, a vertical excitation such as CIS, RPA or TDDFT is required, and the GMH, FCD, FED, FSD, Boys or ER coupling values are calculated based on the excited state results.
Under the two-state approximation, the diabatic reactant and product states are assumed to be a linear combination of the eigenstates. For ET, the choice of such linear combination is determined by a zero transition dipoles (GMH) or maximum charge differences (FCD). In the latter, a donor–acceptor charge difference matrix, , is defined, with elements
where is the matrix element of the density operator between states and .
For EET, a maximum excitation difference is assumed in the FED, in which a excitation difference matrix is similarly defined with elements
where is the sum of attachment and detachment densities for transition , as they correspond to the electron and hole densities in an excitation. In the FSD, a maximum spin difference is used and the corresponding spin difference matrix is defined with its elements as,
where is the spin density, difference between -spin and -spin densities, for transition from .
Since Q-Chem uses a Mulliken population analysis for the integrations in Eqs. (10.15.1.1), (10.15.1.1), and (10.15.1.1), the matrices , and are not symmetric. To obtain a pair of orthogonal states as the diabatic reactant and product states, , and are symmetrized in Q-Chem. Specifically,
The final coupling values are obtained as listed below:
Q-Chem provides the option to control FED, FSD, FCD and GMH calculations after a single-excitation calculation, such as CIS, RPA, TDDFT/TDA and TDDFT. To obtain ET coupling values using GMH (FCD) scheme, one should set $rem variables STS_GMH (STS_FCD) to be TRUE. Similarly, a FED (FSD) calculation is turned on by setting the $rem variable STS_FED (STS_FSD) to be TRUE. In FCD, FED and FSD calculations, the donor and acceptor fragments are defined via the $rem variables STS_DONOR and STS_ACCEPTOR. It is necessary to arrange the atomic order in the $molecule section such that the atoms in the donor (acceptor) fragment is in one consecutive block. The ordering numbers of beginning and ending atoms for the donor and acceptor blocks are included in $rem variables STS_DONOR and STS_ACCEPTOR.
The couplings will be calculated between all choices of excited states with the same spin. In FSD, FCD and GMH calculations, the coupling value between the excited and reference (ground) states will be included, but in FED, the ground state is not included in the analysis. It is important to select excited states properly, according to the distribution of charge or excitation, among other characteristics, such that the coupling obtained can properly describe the electronic coupling of the corresponding process in the two-state approximation.
$molecule 1 1 C 0.679952 0.000000 0.000000 N -0.600337 0.000000 0.000000 H 1.210416 0.940723 0.000000 H 1.210416 -0.940723 0.000000 H -1.131897 -0.866630 0.000000 H -1.131897 0.866630 0.000000 C -5.600337 0.000000 0.000000 C -6.937337 0.000000 0.000000 H -5.034682 0.927055 0.000000 H -5.034682 -0.927055 0.000000 H -7.502992 -0.927055 0.000000 H -7.502992 0.927055 0.000000 $end $rem METHOD CIS BASIS 6-31+G CIS_N_ROOTS 20 CIS_SINGLETS true CIS_TRIPLETS false STS_GMH true !turns on the GMH calculation STS_FCD true !turns on the FCD calculation STS_DONOR 1-6 !define the donor fragment as atoms 1-6 for FCD calc. STS_ACCEPTOR 7-12 !define the acceptor fragment as atoms 7-12 for FCD calc. MEM_STATIC 200 !increase static memory for a CIS job with larger basis set $end
$molecule 0 1 C 0.670518 0.000000 0.000000 H 1.241372 0.927754 0.000000 H 1.241372 -0.927754 0.000000 C -0.670518 0.000000 0.000000 H -1.241372 -0.927754 0.000000 H -1.241372 0.927754 0.000000 C 0.774635 0.000000 4.500000 H 1.323105 0.936763 4.500000 H 1.323105 -0.936763 4.500000 C -0.774635 0.000000 4.500000 H -1.323105 -0.936763 4.500000 H -1.323105 0.936763 4.500000 $end $rem METHOD CIS BASIS 3-21G CIS_N_ROOTS 20 CIS_SINGLETS true CIS_TRIPLETS false STS_FED true STS_DONOR 1-6 STS_ACCEPTOR 7-12 $end
When dealing with multiple charge or electronic excitation centers, diabatic states can be constructed with Boys932 or Edmiston-Ruedenberg928 localization. In this case, we construct diabatic states as linear combinations of adiabatic states with a general rotation matrix that is in size:
The adiabatic states can be produced with any method, in principle, but the Boys/ER-localized diabatization methods have been implemented thus far only for CIS or TDDFT methods in Q-Chem. In analogy to orbital localization, Boys-localized diabatization corresponds to maximizing the charge separation between diabatic state centers:
Here, represents the dipole operator. ER-localized diabatization prescribes maximizing self-interaction energy:
where the density operator at position is
Here, represents the position of the th electron.
These models reflect different assumptions about the interaction of our quantum system with some fictitious external electric field/potential: if we assume a fictitious field that is linear in space, we arrive at Boys localization; if we assume a fictitious potential energy that responds linearly to the charge density of our system, we arrive at ER localization. Note that in the two-state limit, Boys localized diabatization reduces nearly exactly to GMH.932
As written down in Eq. (10.74), Boys localized diabatization applies only to charge transfer, not to energy transfer. Within the context of CIS or TDDFT calculations, one can easily extend Boys localized diabatization931 by separately localizing the occupied and virtual components of , and :
and the occupied/virtual components are defined by
Note that when we maximize the Boys OV function, we are simply performing Boys-localized diabatization separately on the electron attachment and detachment densities.
Finally, for energy transfer, it can be helpful to understand the origin of the diabatic couplings. To that end, we now provide the ability to decompose the diabatic coupling between diabatic states into Coulomb (J), Exchange (K) and one-electron (O) components:1003
$molecule 0 1 he 0 -1.0 1.0 he 0 -1.0 -1.0 he 0 1.0 -1.0 he 0 1.0 1.0 $end $rem METHOD cis CIS_N_ROOTS 4 CIS_SINGLETS false CIS_TRIPLETS true BASIS 6-31g** SCF_CONVERGENCE 8 SYMMETRY false RPA false SYM_IGNORE true LOC_CIS_OV_SEPARATE false ! NOT localizing attachments/detachments separately. ER_CIS_NUMSTATE 4 ! using ER to mix 4 adiabatic states. CIS_DIABATh_DECOMPOSE true ! decompose diabatic couplings into ! Coulomb, exchange, and one-electron components. $end $localized_diabatization On the next line, list which excited adiabatic states we want to mix. 1 2 3 4 $end