**Cite This:**

*J. Chem. Theory Comput.*2014, 10, 9, 3738-3744

# Quasi-diabatic States from Active Space Decomposition

*****E-mail: [email protected] (S. M. Parker).

*****E-mail: [email protected] (T. Shiozaki).

## Abstract

We present *ab initio* theory and efficient algorithms for computing model Hamiltonians of excited-state dynamics in the quasi-diabatic representation. The method is based on a recently developed multiconfiguration electronic structure method, called the active space decomposition method (ASD), in which quasi-diabatic basis states are constructed from physical fragment states. An efficient tree-based algorithm is presented for computing and reusing intermediate tensors appearing in the ASD model. Parallel scalability and wall times are reported to attest the efficiency of our program. Applications to electron, hole, and triplet energy transfers in molecular dimers are presented, demonstrating its versatility.

## 1 Introduction

_{I}|

*d*/

*dR*

_{nuc}|Ψ

_{J}⟩ is zero. The diabatic states allow us to interpret the dynamical processes in an intuitive way. For instance, exciton dynamics in the excited states are described by local states in real space, in which excitons hop between sites. Even though, strictly speaking, diabatic states do not exist in general, (6) there are several approaches that calculate quasi-diabatic states from adiabatic states obtained in quantum chemical calculations: some are based on diagonalization of a property matrix, (7) and others are based on wave function analyses. (8-10) The latter is nevertheless limited to few atom systems, since one needs to connect two geometries by reaction paths on hypersurfaces, a task that is not straightforward for systems containing a few tens of atoms. Diabatization based on the Boys localization has been recently proposed by Subotnik and co-workers. (11, 12)

_{J}

^{*}Π

_{I≠J}Φ

_{I}, where Φ

_{I}and Φ

_{J}

^{*}are the ground and excited states of fragment

*I*and

*J*that are computed by DFT and time-dependent DFT (TD-DFT), respectively. Note that only one-exciton can be described in their model, because of the use of TD-DFT states that are not compact.

## 2 Theory

### Active Space Decomposition

*A*and

*B*, are expressed in the product basis(1)(2)in which orbitals in fragment

*A*and

*B*are assumed to be orthogonal, and |Φ

_{I}

^{A}⟩ are monomer states that are orthogonal with each other:(3)(4)Note that |Ψ

_{IJ}⟩ is properly antisymmetrized with respect to interchanges of electrons in fragment

*A*and

*B*. These orthogonality conditions of orbitals and states are essential to arriving at simple and efficient working equations.

*A*and

*B*, respectively, and (−1)

^{ϕ}is a phase factor due to the rearrangement. The Hamiltonian is, in practice, expressed as a sum of terms with different numbers of operators acting on

*A*and

*B*:(8)The individual terms are, for instance,(9)where orbitals

*j*

_{A}and

*k*

_{A}belong to fragment

*A*, and

*i*

_{B}and

*l*

_{B}belong to

*B*. By regrouping the operators, it can be written as(10)which is in the form of eq 6. Other terms are similarly defined. See ref 30 and its supporting information for details.

*I*or

*I*′ is always the ground state. (22, 24) Hereafter, we adhere to the CAS wave functions, unless otherwise stated.

*ab initio*DMRG code can emulate our ASD model when CAS wave functions are used as a monomer basis. An optimized implementation of ASD, however, allows for adaptation of various active space models for monomer basis, as have been shown elsewhere. (31) Therefore, our model can be seen as a low-entanglement representation (42) of the total wave function using

*a priori*chemical knowledge of the system. The comparisons of ASD and matrix-product state (MPS) ansätze are summarized in Figure 1. The slight empiricism due to manual partitioning of the system is compensated by the potential exactness of the model that allows us to control the accuracy of the decomposition. Among the advantages of ASD over the standard DMRG methods is that one can use configuration-based approximations (such as restricted active space models, or RAS (34)) for monomer states. In this regard, the ASD method is a hybrid of MPS and traditional quantum chemistry.

*A*are orthogonal to all orbitals on fragment

*B*). Next, a set of fragment wave functions are computed using either CAS or RAS fragment wave functions. For each

*type*of fragment state desired (i.e., singlet, triplet, anion, cation, etc.),

*M*states are computed and stored, always with maximum spin projection (

*m*

_{s}=

*S*). The rest of the spin manifold for each type of state (

*m*

_{s}≠

*S*) is computed by applying the spin lowering operator. Finally, transition density matrices between fragment states are computed and the dimer space is diagonalized to find eigenstates.

### Diabatic Model Hamiltonians

^{X}⟩ in eq 2 are eigenstates of the spin angular momentum operator, (

*Ŝ*

^{X})

^{2}, the spin-projection operator,

*Ŝ*

_{z}

^{X}, and the electron number operator,

*N̂*

^{X}, all confined to act on monomer

*X*. Therefore, we can straightforwardly define diabatic subspaces as sets of dimer basis states in which monomer

*A*and

*B*have the same charge (

*q*

_{A}and

*q*

_{B}) and spin (

*S*

_{A}and

*S*

_{B}). Then, diabatic model states are found by diagonalizing the diabatic sub-block of the Hamiltonian, i.e., by diagonalizing(11)where

*P̂*

_{diabatic}is the projector onto the quasi-diabatic states of interest.

_{I,q,S}

^{X}⟩ is the

*I*th eigenstate of monomer

*X*that has charge

*q*and spin

*S*. To find |Ψ

_{0+}⟩, we compute the lowest lying eigenstates of the subspace spanned by the products of all neutral singlet monomer

*A*states and all cationic doublet monomer

*B*states, {|Φ

_{I,0,0}

^{A}⟩} ⊗ {|Φ

_{J,+1,1/2}

^{B}⟩}. |Ψ

_{+0}⟩ is computed similarly. The model Hamiltonian for the direct electron transfer is then computed as(14)For mediated processes an effective model Hamiltonian must be formed. Examples of such processes include triplet energy transfer (43) and singlet fission (44) both of which are mediated by charge-transfer states. Appealing to the symmetrized Bloch equation (45) (or, more specifically, to quasi-degenerate perturbation theory (46) to second order), one arrives at(15)where the sum is over select (or, possibly, all) dimer states that are not in the diabatic spaces.

## 3 Algorithms and Implementations

### Tree-Based Algorithm for Γ Intermediates

_{I}⟩) are collected into a single tree in which said ket vector forms the root node. All other nodes (i.e., nonroot nodes) contain

*sets*of bra vectors (e.g., {⟨Φ

_{J}|}). An individual tensor element is expressed as a path between the root node and some other node: starting from the ket vector (root node), each edge applies an operator and the terminal node (of the path) applies a bra vector; that is, an inner product is taken with the CI vectors at the terminal node of the path. Since, at most, three operators need to be applied, the maximum height of the tree is 4. Intermediate nodes along the path have no physical meaning; only the root node and the terminal node of the path have meaning.

*types*of edges (operators) that are needed. There are four types of edges in the skeletons, one for each type of second quantization operator: (α)

^{†}, (β)

^{†}, (α), and (β). The full trees are generated from the skeletons by replacing each skeleton edge with one real edge per active orbital. Thus, the pair of skeleton edges (α)

^{†}(α) represents

*n*

_{act}

^{2}real edges where

*n*

_{act}is the number of active orbitals on the monomer. A sample skeleton tree is shown in Figure 3.

_{0}⟩ = |Φ

_{I}⟩, where |Φ

_{I}⟩ is the ket vector associated with the root of the current tree. Furthermore, each MPI process stores a local copy of each transition density matrix, Γ

_{ξ}

^{IJ,[n]}, where, hereafter, [

*n*] denotes MPI process

*n*.

*I*′⟩ = (−1)

^{ϕ}

*â*|

*I*⟩, where |

*I*⟩ and |

*I*′⟩ are Slater determinants] are computed using bit-wise operations through the bitset class of the C++ standard. Then, inner products with all CI vectors contained in the node are computed and stored in the appropriate (local) transition density matrix. Since the bra vectors held at each node of the tree are distributed across MPI processes, the inner product is taken within each MPI process,(16)and later summed across all the processes as(17)The standard MPI_Allreduce is used for summation.

_{I}⟩ is replicated and stored locally as |Ξ

_{0}⟩, and the first intermediate(20)is computed. The inner product with the locally available portion of the bra vector, ⟨Φ

_{j}

^{[n]}|, is then formed,(21)Next, using the above |Ξ

_{1}⟩ intermediate, the second intermediate |Ξ

_{2}⟩ and Γ

_{pα,qα}

^{IJ′,[n]}are likewise computed as(22)(23)Finally, transition density matrices are summed as in eq 17. For a more-detailed description of the algorithm for traversal of the Γ trees, see Figure 4.

### Parallel Algorithms and Timing

*rs*) with γ

_{begin}≤ γ ≤ γ

_{end}. In our code, γ

_{begin}and γ

_{end}on each MPI process is chosen to be at the basis shell boundaries. Here {···} denotes that the code is parallelized using these indices. The Fock matrix is computed by the standard algorithm,(24)(25)(26)(27)(28)(29)(30)with . The half index transformation (eq 24) and multiplication of in eq 26 only involves intranode operations and is threaded via the underlying BLAS calls. Similarly, computations of the exchange builds (eq 28) and the coulomb builds (eqs 29 and 30) are performed without communications, and the resulting two-index matrices are summed. The most communication-intensive steps are eqs 25 and 27, in which we need to collectively transpose the distributed three-index half-transformed integrals.

*M*monomer states are computed simultaneously in each diabatic subspace using Davidson diagonalization. The Davidson iteration requires one to compute(31)for each state

*m*in the quasi-diabatic sector

*X*(

*Ê*

_{rs}and

*Ê*

_{rs,tu}are standard spin-free operators). Since, in our examples,

*M*is greater than or equal to the number of MPI processes, this task can be trivially parallelized. The total wave functions are distributed to all the MPI processes; therefore, before each Davidson iteration, the

*c*

^{X,m}are localized onto all the MPI processes. The

*M*σ formations are split among all the MPI processes and each σ evaluation is performed locally. Our current implementation is sequential in

*X*to minimize memory usage, making it scalable up to

*M*MPI processes. For CAS wave functions, the evaluation of the two electron part of eq 31 is based on the algorithm proposed by Harrison and Zarrabian using

*N*

_{elec}– 2 electron intermediate determinants. (47) For RAS wave functions, we use the procedure proposed by Olsen et al. (34)

## 4 Numerical Examples

*M*, in which they are expanded. The “dimer space” refers to the full set of dimer states, i.e., the products of all charge- and spin-allowed combinations of monomer states, and may include types of monomer states not included in the model spaces. The portion of the dimer space not included in the model space is used only to compute mediated couplings.

*H*

_{if}. As a first example, we explore the distance dependence of these coupling matrix elements for several elementary electronic processes in a stacked benzene dimer: electron transfer (ET), hole transfer (HT), and triplet excitation energy transfer (TT). These processes are summarized in Figure 2.

_{±0}⟩ and |Ψ

_{0±}⟩. The four model states are thus |Ψ

_{±0}

^{0}⟩, |Ψ

_{±0}

^{1}⟩, |Ψ

_{0±}

^{0}⟩ and |Ψ

_{0±}

^{1}⟩, where the superscript labels the two degenerate solutions with the same diabatic character. We use the def2-TZVPP basis set (48) and

*M*= 32 monomer states using CAS(6 –

*q*,6) wave functions, where

*q*is the charge of the monomer. The resulting model Hamiltonian for ET in units of eV at a separation of 4.0 Å is(32)This value for the coupling is similar to, but lower than, the value of 570 meV found by You et al., (49) which was derived from the splittings of adiabatic states computed with spin-flip CI singles (estimated from Figure 4 of ref 49). Exploiting a property of symmetric homodimers, couplings can be inferred from adiabatic energy gaps through Δ

*E*= 2

*H*

_{if}. The model Hamiltonian for HT at a separation of 4.0 Å is(33)This compares well with HT couplings of 214 meV from Kubas et al. derived from the splittings of adiabatic states computed with NEVPT2. (50)

_{+ –}⟩ and |Ψ

_{– +}⟩ in the summation in eq 15. Then, the mediated model Hamiltonian for TT at 4.0 Å is(35)The magnitudes of all the coupling elements as a function of separation are shown in Figure 6.

*q*,6) for the benzene monomer states and CAS(10 –

*q*,10) for the naphthalene monomer states, all with

*M*= 32. The resulting mediated Hamiltonian is(36)where |Ψ̃

_{B↑N0}⟩ and |Ψ̃

_{B0N↑}⟩ denote the states in which the triplet resides on the benzene and the naphthalene, respectively. Similarly, our recently reported model Hamiltonians for singlet fission in tetracene and pentacene dimers built from experimental crystal structures demonstrates that ASD can efficiently treat asymmetric dimers. (31)

## 5 Conclusions

## Acknowledgment

This work has been supported by Office of Basic Energy Sciences, U.S. Department of Energy (Grant No. DE-FG02-13ER16398). S.M.P. thanks Profs. Mark Ratner and Tamar Seideman for encouragement.

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Quantities investigated were atomization energies, dipole moments and structure parameters for Hartree-Fock, d. functional theory and correlated methods, for which we had chosen Moller-Plesset perturbation theory as an example. Finally recommendations are given which type of basis set is used best for a certain level of theory and a desired quality of results.**49**You, Z.-Q.; Shao, Y.; Hsu, C.-P. Chem. Phys. Lett. 2004, 390, 116– 123Google ScholarThere is no corresponding record for this reference.**50**Kubas, A.; Hoffmann, F.; Heck, A.; Oberhofer, H.; Elstner, M.; Blumberger, J. J. Chem. Phys. 2014, 140, 104105Google ScholarThere is no corresponding record for this reference.**51**Closs, G. L.; Johnson, M. D.; Miller, J. R.; Piotrowiak, P. J. Am. Chem. Soc. 1989, 111, 3751– 3753Google Scholar51https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXitlSrsL8%253D&md5=3a5c42885c012c407dea59752fbc0892A connection between intramolecular long-range electron, hole, and triplet energy transfersCloss, Gerhard L.; Johnson, Mark D.; Miller, John R.; Piotrowiak, PiotrJournal of the American Chemical Society (1989), 111 (10), 3751-3CODEN: JACSAT; ISSN:0002-7863.Frontier MO theory predicts a simple relationship between intramol. electron transfer (ET), hole transfer (HT), and triplet energy transfer (TT). Within this simplified framework TT is the simultaneous exchange of two electrons between donor and acceptor, one between the LUMO's and the other between HOMO's. Consequently, the probability of TT should be proportional to the product of the probabilities of ET (exchange among LUMO's) and HT (exchange among HOMO's). This prediction is restricted to the influence of the electronic coupling on the rate and requires the Franck Condon factors to be const. throughout the series. Exptl. data on donor acceptor pairs, covalently linked by ten different spacers, are presented for ET, HT, and TT. The donors are 4-biphenylyl for ET and HT and 4-benzophenoneyl for TT. The acceptors are 2-naphthyl in all three series. The spacers are decalins and cyclohexanes with different regio- and stereochem. attachments. The prediction is reasonably well obeyed. A better correlation is obtained, however, when corrections are made for the change of the Franck Condon factors arising from the distance dependence of the reorganization energy in ET and HT reactions.**52**Dexter, D. L. J. Chem. Phys. 1953, 21, 836– 850Google Scholar52https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaG3sXks12gtg%253D%253D&md5=a5105d74d45df5a63d99cb6f4aa1d8b4A theory of sensitized luminescence in solidsDexter, D. L.Journal of Chemical Physics (1953), 21 (), 836-50CODEN: JCPSA6; ISSN:0021-9606.The term "sensitized luminescence" in cryst. phosphors refers to the phenomenon whereby an impurity (activator or emitter) is enabled to luminesce upon the absorption of light in a different type of center (sensitizer or absorber) and upon the subsequent radiationless transfer of energy from the sensitizer to the activator. The resonance theory of F.ovrddot.orster (C.A. 43 6079i) which involves only allowed transitions, is extended to include transfer by means of forbidden transitions which, it is concluded, are responsible for the transfer in all inorg. systems yet investigated. The transfer mechanisms of importance are, in order of decreasing strength, the overlapping of the elec. dipole fields of the sensitizer and the activator, the overlapping of the dipole field of the sensitizer with the quadrupole field of the activator, and exchange effects. These mechanisms will give rise to "sensitization" of about 103-104, 102, and 30 lattice sites surrounding each sensitizer in typical systems. The dependence of transfer efficiency upon sensitizer and activator concn. and on temp. are discussed. Application is made of the theory to exptl. results on inorg. phosphors, and expts. are suggested.

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**1**Helgaker, T.; Jørgensen, P.; Olsen, J. Molecular Electronic-Structure Theory; Wiley: West Sussex, U.K., 2000.There is no corresponding record for this reference.**2**Christiansen, O.; Jørgensen, P.; Hättig, C. Int. J. Quantum Chem. 1998, 68, 1– 522https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaK1cXisVWiu7s%253D&md5=c3e78f81cacc2f2920cd1f402a2a539aResponse functions from Fourier component variational perturbation theory applied to a time-averaged quasienergyChristiansen, Ove; Jorgensen, Poul; Hattig, ChristofInternational Journal of Quantum Chemistry (1998), 68 (1), 1-52CODEN: IJQCB2; ISSN:0020-7608. (John Wiley & Sons, Inc.)Frequency-dependent response functions can conveniently be derived from the time-averaged quasienergy. The variational criteria for the quasienergy dets. the time-evolution of the wave-function parameters and the time-averaged time-dependent Hellmann-Feynman theorem allows an identification of response functions as derivs. of the quasienergy. The quasienergy therefore plays the same role as the usual energy in time-independent theory, and the same techniques can be used to obtain computationally tractable expressions for response properties, as for energy derivs. in time-independent theory. This includes the use of the variational Lagrangian technique for obtaining expressions for mol. properties in accord with the 2n+1 and 2n+2 rules. The derivation of frequency-dependent response properties becomes a simple extension of the variational perturbation theory to a Fourier component variational perturbation theory. The generality and simplicity of this approach are illustrated by derivation of linear and higher-order response functions for both exact and approx. wave functions and for both variational and nonvariational wave functions. Examples of approx. models discussed in this article are coupled-cluster, SCF, and second-order Moller-Plesset perturbation theory. A discussion of symmetry properties of the response functions and their relation to mol. properties is also given, with special attention to the calcn. of transition- and excited-state properties.**3**Riplinger, C.; Neese, F. J. Chem. Phys. 2013, 138, 0341063https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3sXpslOqtw%253D%253D&md5=4327115b95524107245acb44ff4aaa7bAn efficient and near linear scaling pair natural orbital based local coupled cluster methodRiplinger, Christoph; Neese, FrankJournal of Chemical Physics (2013), 138 (3), 034106/1-034106/18CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)In previous publications, it was shown that an efficient local coupled cluster method with single- and double excitations can be based on the concept of pair natural orbitals (PNOs) . The resulting local pair natural orbital-coupled-cluster single double (LPNO-CCSD) method has since been proven to be highly reliable and efficient. For large mols., the no. of amplitudes to be detd. is reduced by a factor of 105-106 relative to a canonical CCSD calcn. on the same system with the same basis set. In the original method, the PNOs were expanded in the set of canonical virtual orbitals and single excitations were not truncated. This led to a no. of fifth order scaling steps that eventually rendered the method computationally expensive for large mols. (e.g., >100 atoms). In the present work, these limitations are overcome by a complete redesign of the LPNO-CCSD method. The new method is based on the combination of the concepts of PNOs and projected AOs (PAOs). Thus, each PNO is expanded in a set of PAOs that in turn belong to a given electron pair specific domain. In this way, it is possible to fully exploit locality while maintaining the extremely high compactness of the original LPNO-CCSD wavefunction. No terms are dropped from the CCSD equations and domains are chosen conservatively. The correlation energy loss due to the domains remains below <0.05%, which implies typically 15-20 but occasionally up to 30 atoms per domain on av. The new method has been given the acronym DLPNO-CCSD ("domain based LPNO-CCSD"). The method is nearly linear scaling with respect to system size. The original LPNO-CCSD method had three adjustable truncation thresholds that were chosen conservatively and do not need to be changed for actual applications. In the present treatment, no addnl. truncation parameters have been introduced. Any addnl. truncation is performed on the basis of the three original thresholds. There are no real-space cutoffs. Single excitations are truncated using singles-specific natural orbitals. Pairs are prescreened according to a multipole expansion of a pair correlation energy est. based on local orbital specific virtual orbitals (LOSVs). Like its LPNO-CCSD predecessor, the method is completely of black box character and does not require any user adjustments. It is shown here that DLPNO-CCSD is as accurate as LPNO-CCSD while leading to computational savings exceeding one order of magnitude for larger systems. The largest calcns. reported here featured >8800 basis functions and >450 atoms. In all larger test calcns. done so far, the LPNO-CCSD step took less time than the preceding Hartree-Fock calcn., provided no approxns. have been introduced in the latter. Thus, based on the present development reliable CCSD calcns. on large mols. with unprecedented efficiency and accuracy are realized. (c) 2013 American Institute of Physics.**4**Nakatsuji, H. Acc. Chem. 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Phys. 1993, 99, 7983– 7992There is no corresponding record for this reference.**47**Harrison, R. J.; Zarrabian, S. Chem. Phys. Lett. 1989, 158, 393– 398There is no corresponding record for this reference.**48**Weigend, F.; Ahlrichs, R. Phys. Chem. Chem. Phys. 2005, 7, 3297– 330548https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BD2MXpsFWgu7o%253D&md5=a820fb6055c993b50c405ba0fc62b194Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracyWeigend, Florian; Ahlrichs, ReinhartPhysical Chemistry Chemical Physics (2005), 7 (18), 3297-3305CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)Gaussian basis sets of quadruple zeta valence quality for Rb-Rn are presented, as well as bases of split valence and triple zeta valence quality for H-Rn. The latter were obtained by (partly) modifying bases developed previously. A large set of more than 300 mols. representing (nearly) all elements-except lanthanides-in their common oxidn. states was used to assess the quality of the bases all across the periodic table. Quantities investigated were atomization energies, dipole moments and structure parameters for Hartree-Fock, d. functional theory and correlated methods, for which we had chosen Moller-Plesset perturbation theory as an example. Finally recommendations are given which type of basis set is used best for a certain level of theory and a desired quality of results.**49**You, Z.-Q.; Shao, Y.; Hsu, C.-P. Chem. Phys. Lett. 2004, 390, 116– 123There is no corresponding record for this reference.**50**Kubas, A.; Hoffmann, F.; Heck, A.; Oberhofer, H.; Elstner, M.; Blumberger, J. J. Chem. Phys. 2014, 140, 104105There is no corresponding record for this reference.**51**Closs, G. L.; Johnson, M. D.; Miller, J. R.; Piotrowiak, P. J. Am. Chem. Soc. 1989, 111, 3751– 375351https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaL1MXitlSrsL8%253D&md5=3a5c42885c012c407dea59752fbc0892A connection between intramolecular long-range electron, hole, and triplet energy transfersCloss, Gerhard L.; Johnson, Mark D.; Miller, John R.; Piotrowiak, PiotrJournal of the American Chemical Society (1989), 111 (10), 3751-3CODEN: JACSAT; ISSN:0002-7863.Frontier MO theory predicts a simple relationship between intramol. electron transfer (ET), hole transfer (HT), and triplet energy transfer (TT). Within this simplified framework TT is the simultaneous exchange of two electrons between donor and acceptor, one between the LUMO's and the other between HOMO's. Consequently, the probability of TT should be proportional to the product of the probabilities of ET (exchange among LUMO's) and HT (exchange among HOMO's). This prediction is restricted to the influence of the electronic coupling on the rate and requires the Franck Condon factors to be const. throughout the series. Exptl. data on donor acceptor pairs, covalently linked by ten different spacers, are presented for ET, HT, and TT. The donors are 4-biphenylyl for ET and HT and 4-benzophenoneyl for TT. The acceptors are 2-naphthyl in all three series. The spacers are decalins and cyclohexanes with different regio- and stereochem. attachments. The prediction is reasonably well obeyed. A better correlation is obtained, however, when corrections are made for the change of the Franck Condon factors arising from the distance dependence of the reorganization energy in ET and HT reactions.**52**Dexter, D. L. J. Chem. Phys. 1953, 21, 836– 85052https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADyaG3sXks12gtg%253D%253D&md5=a5105d74d45df5a63d99cb6f4aa1d8b4A theory of sensitized luminescence in solidsDexter, D. L.Journal of Chemical Physics (1953), 21 (), 836-50CODEN: JCPSA6; ISSN:0021-9606.The term "sensitized luminescence" in cryst. phosphors refers to the phenomenon whereby an impurity (activator or emitter) is enabled to luminesce upon the absorption of light in a different type of center (sensitizer or absorber) and upon the subsequent radiationless transfer of energy from the sensitizer to the activator. The resonance theory of F.ovrddot.orster (C.A. 43 6079i) which involves only allowed transitions, is extended to include transfer by means of forbidden transitions which, it is concluded, are responsible for the transfer in all inorg. systems yet investigated. The transfer mechanisms of importance are, in order of decreasing strength, the overlapping of the elec. dipole fields of the sensitizer and the activator, the overlapping of the dipole field of the sensitizer with the quadrupole field of the activator, and exchange effects. These mechanisms will give rise to "sensitization" of about 103-104, 102, and 30 lattice sites surrounding each sensitizer in typical systems. The dependence of transfer efficiency upon sensitizer and activator concn. and on temp. are discussed. Application is made of the theory to exptl. results on inorg. phosphors, and expts. are suggested.