State-Interaction Pair-Density Functional Theory Can Accurately

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State-Interaction Pair-Density Functional Theory Can Accurately Describe a Spiro Mixed Valence Compound Sijia S. Dong, Kevin Benchen Huang, Laura Gagliardi, and Donald G. Truhlar J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b01301 • Publication Date (Web): 19 Feb 2019 Downloaded from http://pubs.acs.org on February 23, 2019

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The Journal of Physical Chemistry

State-Interaction Pair-Density Functional Theory Can Accurately Describe a Spiro Mixed Valence Compound 1

Sijia S. Dong,a† Kevin Benchen Huang,a,b† Laura Gagliardi,a* and Donald G. Truhlara* a

Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, MN 55455, USA b

College of Chemistry, Nankai University, Tianjin, China, 300071

Abstract Mixed-valence compounds with strong couplings between electronic states constitute one of the most challenging types of multireference systems for electronic structure theory. Previous work on a model mixed-valence compound, the 2,2´,6,6´-tetrahydro-4H,4´H-5,5´spirobi[cyclopenta[c]pyrrole] cation, showed that multireference perturbation theory (MRPT) can give a physical energy surface for the mixed-valence compound only by going to the third order or by using a scheme involving averaging orbital energies in a way specific to mixedvalence systems. In this study, we show that second-order MRPT methods (CASPT2, MSCASPT2, and XMS-CASPT2) can give good results by calculating the Fock operator for the zeroth-order Hamiltonian using the state-averaged density matrix. We also show that stateinteraction pair-density functional theory (SI-PDFT) is free from the unphysical behavior of previously tested second-order MRPT methods for this prototype mixed-valence compound near the avoided crossing. This is very encouraging because of the much lower cost in applying SIPDFT to large or complex systems.

1

Current address: Institute for Molecular Engineering and Materials Science Division, Argonne National Laboratory, Lemont, IL 60439, USA

ACS Paragon Plus Environment

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Introduction Charge transfer (CT) is a common chemical phenomenon and is critical in modern technology.1-7 To simulate CT processes quantitatively, the first step is an accurate description of the potential energy surface of the CT system. Among CT systems, mixed-valence compounds with strong state coupling are one of the most difficult types to simulate. Mixed-valence compounds are generally composed of two organic moieties or two metal centers with the property that they can support intramolecular charge transfer or delocalized wave functions with non-integer oxidation numbers on atoms of two identical or similar subsystems. A popular classification scheme is that of Robin and Day8 by which mixed-valence compounds can be classified into three categories, from completely localized (Type I) to fully delocalized (Type III). It is the intermediate case (Type II), where the weak electronic interaction between the two moieties causes partial delocalization of the odd electron, that are both the most interesting for potential new applications and the most challenging for fundamental valence theory. In the present study, we consider the cation of the widely studied 2,2´,6,6´-tetrahydro4H,4´H-5,5´-spirobi[cyclopenta[c]pyrrole] molecule (Figure 1), which is a Type II system, and which will simply be called the Spiro cation. The electronic states of the Spiro cation have been used as a prototype in a number of studies to investigate how well electronic structure theory methods can simulate CT processes. The equilibrium geometry of the neutral Spiro molecule has D2d symmetry and consists of two conjugated π systems that are perpendicular to each other and linked by a 𝜎 bridge with a single carbon atom shared by the two ring systems. The equilibrium structure of the cation (i.e., its lowest-energy structure) has the positive charge on one or another of the cyclopenta[c]pyrrole rings (moieties), and the point group of the molecule is thereby reduced to C2v. In this work, we focus on the two lowest electronic states, 12A2 and 22A2 in C2v symmetry, and we consider the charge (hole) transfer reaction from one ring system to the other


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ring system. Our primary purpose is to demonstrate that state-interaction pair-density functional theory (SI-PDFT)9 is effective and efficient in describing charge transfer in this prototype mixedvalence compound. We will also show that, unlike the conclusion from previously published work (summarized below) that multireference second-order perturbation theory (MRPT2) needs to be modified for the mixed-valence system to give reasonable results, standard MRPT2 calculations are sufficient to give good results for this system.

Figure 1. The atomic numbering of the Spiro cation. Theoretical Methods Charge-Transfer Reaction Path Using Multireference Perturbation Theory (MRPT) The CT process of the Spiro cation involves the charge migration from one ring system to the other ring system. Midway through this kind of process, depending on the path taken, there is a crossing (conical intersection) or locally avoided crossing (passage to the side of a conical intersection) of the two lowest potential energy surfaces such as what occurs in the Marcus theory of electron transfer.10 This process has been extensively studied during the last several decades using a variety of theoretical methods,11 including complete active space self-consistent field (CASSCF),12 complete active space second-order perturbation theory (CASPT2),13 and nelectron valence state perturbation theory (NEVPT),14-17 with disparate results. Strikingly, it was



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found that MRPT2 methods, in particular CASPT2 and NEVPT2, failed to account for the behavior of the potential energy surface near the avoided crossing,18 yielding an unphysical well near the avoided crossing. The well was shown to be due to a sudden modification of the canonical orbital energies in the denominator of the perturbation theory energy expression in the vicinity of the avoided crossing. By going to the third-order energy correction, error cancellation causes the total energy to become approximately correct, even though neither the second- nor the third-order perturbation energy term alone has the right behavior.18 In the same and a later work,11 Pastore et al. showed that, using a new method called NEVPT2(av) in which for each geometry along the reaction path an average energy between the orbitals of the first two electronic states of the same symmetry is used for NEVPT2, this problem can also be solved. In the present work, we find that standard MRPT2 methods (i.e., MRPT2 methods that are not modified for the Spiro problem such as in the modifications of Pastore et al.) can give good results, in particular we obtain good results by using CASPT2 and multistate CASPT2 (MSCASPT2) with the state-averaged density matrix for calculating the Fock operator for the zerothorder Hamiltonian (H0). In this article, we use “ss” to denote CASPT2 or MS-CASPT2 calculations with H0 calculated from the density from each state separately, and we use “sa” to denote CASPT2 or MS-CASPT2 calculations with H0 calculated from the state-averaged density matrix. Note that MS-CASPT2(ss) is the original MS-CASPT2 method19 normally simply called MS-CASPT2, but called SS-SR-CASPT2 in Molpro.20 MS-CASPT2(sa) is an attractive alternative;21 it is called MS-MR-CASPT2 in Molpro. Extended MS-CASPT2 (XMS-CASPT2) is a method22, 23 that improves upon MS-CASPT2 and makes the reference states invariant to unitary rotations; it was developed to improve the behavior of the theory near avoided crossings and conical intersections. In the present work, we


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also show that good results can be obtained by XMS-CASPT2 using the state-averaged density matrix when calculating the Fock operator. XMS-CASPT2 is XMS-CASPT2(sa) by definition, and there is no XMS-CASPT2(ss); therefore, we need not include a parenthetical specification when we discuss XMS-CASPT2. State-Interaction Pair-Density Functional Theory One deterrent to using MRPT2 or higher-order perturbation theory methods for large or complex systems is their computational cost. Previously, we developed the multiconfiguration pair-density functional theory (MC-PDFT) as an alternative method for efficient simulation of multireference systems.24 MC-PDFT calculates the kinetic energy and the classical Coulomb energy from a multiconfigurational reference wave function, and calculates the rest of the electronic energy from an on-top density functional which is a functional of the density and ontop pair density calculated from the multiconfigurational reference wave function. MC-PDFT is similar to other multireference methods such as CASPT2 in that it calculates a multiconfigurational reference wave function, for example by an MCSCF method, but it differs from pure wave function methods in that the final energy calculation includes a component calculated using an on-top functional, and it is less computationally expensive than MRPT methods. The SI-PDFT method is an extension of MC-PDFT developed to treat strongly coupled states, i.e., nearly degenerate states;9 SI-PDFT stands in relation to MC-PDFT in an analogous way to the relationship of MS-CASPT2 to CASPT2. The SI-PDFT method is implemented in OpenMolcas.25, 26 In the first step, SI-PDFT involves two calculations, one to generate the state-specific CASSCF ground state and another to generate the first N state-averaged CASSCF states; we will use the abbreviation SS-CASSCF for the former calculation and the abbreviation SA-CASSCF



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for the latter calculation. In the second step, one obtains a new set of states as follows.9 The new ground state is Θ# = 𝐴#

* '+#

𝜓'() 𝜓'() 𝜓 (( ,

(1)

where 𝐴# is a normalization constant, and the excited states Θ- are Θ- = 𝐴- 𝜓-() −

-0# '+#

Θ' Θ' 𝜓/() ,

j>1

(2)

where Aj is a normalization constant. Equation (Eq.) 1 projects the SS-CASSCF ground state into the space spanned by the N SA-CASSCF states, and Eq. 2 is obtained by Gram-Schmidt orthogonalization. In the third step, one constructs an N ´ N model-space Hamiltonian matrix. The diagonal elements are computed from the Θ- functions by MC-PDFT, and the off-diagonal elements are computed by wave function theory, i.e., 𝐻'- = Θ' 𝐻 Θ- ,

i≠j

(3)

The eigenvalues of this Hamiltonian matrix are the final SI-PDFT energies. Computational Details The neutral Spiro molecule has D2d symmetry, and its cation has C2v symmetry. At the ground state, the equilibrium geometries of the two moieties of the cation are slightly different, indicating the charge distribution is different for the two moieties. For convenience, in the figures of this article we orient the Spiro cation such that one ring system is on the left and the other ring system is on the right. The C2v equilibrium structure with the hole localized on the left is denoted A, and the C2v equilibrium structure with the hole localized on the right is denoted B. The reaction path for the Spiro cation with the hole migrating from one side to the other (“left to right” is equivalent to “right to left”) is calculated by the linear synchronous transit27 method. It has been pointed out that reaction path obtained in this way is very close to the optimal one.28


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First the geometries of A and B are optimized using the restricted-open-shell Hartree-Fock (ROHF) method with the TZP basis set29 in Gaussian 09.30 We used ROHF to be consistent with previous studies of the Spiro cation reaction path.11, 18, 28 It has been suggested that the geometry depends little on the method used and that bond lengths calculated by ROHF and CASSCF(7,4) with the ANO-L basis set31 with TZP contraction level (3s1p for H and 4s3p1d for C and N) differ by within 0.003 Å.28 The TZP basis set we used has 3s2p1d for H and 5s3p2d1f for C and N, and the optimized bond lengths differ from the CASSCF(7,4)/ANO-L optimized bond lengths by only 0.001 - 0.004 Å. The CM5 charge was calculated with ROHF/TZP. Then the path is parametrized by a unitless progress variable x running from -1.5 to 1.5 (with a step size of Dx = 0.05), and the Cartesian coordinates 𝑄3 (where g = 1, 2, …, 3Natoms) of points on the path are given by Q3 𝜉 =

# 6

− 𝜉 𝑄3𝐀 +

# 6

+ 𝜉 𝑄3𝐁 ,

(4)

The two local minima are located where x = -0.50 (A) and x = 0.50 (B). The Cartesian coordinates of 𝑄) are in Table S1 (tables with a prefix S are in the Supporting Information (SI)). When x = 0, the geometry is the average of those at the two local minima, and it has D2d symmetry; this may be considered an approximation to the transition structure for charge transfer. We carried out CASSCF and SI-PDFT calculations using OpenMolcas.25, 26 All SI-PDFT calculations are with the tPBE functional. The CASPT2, MS-CASPT2, and XMS-CASPT2 calculations were carried out using Molpro.20 Due to the size of the molecule, all the 1s and 2s atomic orbitals of nitrogen and carbon atoms are frozen in the MS-CASPT2 and XMS-CASPT2 calculations. The 1s and 2s atomic orbitals of nitrogen and carbon atoms are also frozen in CASPT2 calculations unless otherwise noted. No level shift was used. To obtain the CASSCF wave function, we scanned the potential energy curve from 𝜉 = 0 to 𝜉 = -1.5, where the guess



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orbital of SA- and SS-CASSCF of the first geometry was taken from unrestricted HF, and the guess orbital of each subsequent geometry was taken from the optimized SA- and SS-CASSCF natural orbitals of the previous point. The active space of CASSCF calculations is defined by (ne,no) where ne is the number of active electrons and no is the number of active orbitals. We tested the (7,4) and (11,10) active spaces in CASSCF calculations using the 6-31G(d) basis set for the energy calculations of the cation. For the neutral Spiro molecule, the restrictedHF/TZP orbitals from HOMO-4 to HOMO are 12b1, 12b2, 3a2, and 4a2, respectively, and are π orbitals. For the cation, one of the two a2 orbitals is singly occupied, and we choose these four π orbitals as the active orbitals of the (7,4) active space. The 4 lowest states generated from this active space are of A2, A2, B1, B2 symmetries. Here, we focus on the calculation of the two 2A2 states, which are referred to as the ground state and the excited state in this article. For the (7,4) active space, for states with A2 symmetry, there are only two CSFs that contribute to the wave function, each CSF corresponding to the configuration with one electron singly occupied in either a2 orbital. We also tested the active space (11,10) in which the active orbitals are all the valence π orbitals. For the (11,10) active space, a high contraction level21 needs to be used in CASPT2 calculations to make the calculation affordable, but MS-CASPT2 and XMS-CASPT2 are not implemented for this contraction level in Molpro. Evaluation of the effect of contraction levels on the potential energy curves is detailed in Section 1 of the SI. Results and Discussion CASSCF Reference Wave Function for the Spiro Cation The charge distribution of the Spiro cation is shown in Figure 2. The moiety having more positive charge is labeled [+], and the other moiety is labeled [n]. The difference in bond lengths between the two moieties of the Spiro cation are reported in Table S2, which shows that [+] has


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longer C4-C5 bond and shorter C4-C3 and C5-N1 bonds compared to [n]. This is consistent with the picture that for the pyrrole ring in [+], a double bond is formed between C3 and C4, and positive charge is delocalized around C5-N1-C2, as shown in Figure 2. CM5 charge32 supports this picture, where the CM5 charge on C2 and C5 in [+] is 0.18 while that in [n] is only 0.00.

Figure 2. Illustration of the pyrrole ring in the [+] moiety and the [n] moiety of the Spiro cation (structure A) and its CM5 charge calculated with ROHF/TZP To investigate the character of the CASSCF wave function along the reaction path, we analyze the dominant configurations, their configuration interaction (CI) coefficients, natural orbitals, and natural orbital occupation numbers of selected geometries (Tables S3-S5). The dominant configuration of the ground state for both the (7,4) and (11,10) active spaces has the unpaired electron in the 4a2 orbital, as shown in Figure 3. For the 22A2 state, the dominant configuration has the unpaired electron in the 3a2 orbital instead. For C2v geometries, the 4a2 orbital is localized on the [+] moiety and the 3a2 orbital is localized on the [n] moiety, while both orbitals for the D2d geometry are distributed on both moieties. The weight (square of the CI coefficient) of the dominant configuration remains between 0.833 and 0.867 for the reaction path,



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but the natural orbitals shown in Figure 3 and Tables S4-S5 demonstrate the charge changes from localized on one moiety to equally distributed on both moieties and then move to the other moiety. The CI coefficients of the Spiro molecule implies some multireference character of the system, but CI coefficients do not provide a straightforward measure of the multireference character. To further characterize the multireference character of the Spiro cation, we used the multireference diagnostic,33 which is defined by 𝑀=

1 2 − 𝑛MCDONO + 2

𝑛-(SOMO) − 1 + 𝑛MCUNO -(SOMO)

where nMCDONO is the natural orbital occupation number of the most-correlated doubly-occupied orbital (MCDONO), nj(SOMO) is the natural orbital occupation number of the j-th singly-occupied orbital (SOMO), and nMCDONO is the natural orbital occupation number of the most-correlated unoccupied orbital (MCUNO). When M < 0.05, the multireference character is classified as small. When 0.05 ≤ M ≤ 0.10, the multireference character is classified as modest. When M > 0.10, the multireference character is classified as large. For the 12A2 state, as the charge moves from being localized on one moiety at ξ = -1.5 to being equally distributed on both moieties at ξ = 0, the M value increases from 0.064 to 0.077, as displayed in Table S4. For the 22A2 state, the M value decreases from 0.092 at ξ = -1.5 to 0.080 at ξ = 0 instead (Table S5). This demonstrates that the Spiro molecule has moderate multireference character for the two states of interest. As the charge moves from one moiety to the other, the multireference character first increases and then decreases for the ground state, and first decreases and then increases for the 22A2 state. This implies that the challenge in computing the electronic structure of this system not only lies in the


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near-degeneracy of its frontier orbitals (3a2 and 4a2), but also comes from its multireference nature.

Figure 3. Selected SA(2)-CASSCF(11,10) natural orbitals of the 12A2 state and their occupation numbers for a C2v geometry (𝜉 = –0.5) and the D2d geometry (𝜉 = 0). Comparison of CASSCF, SI-PDFT, and MRPT Reaction Paths To compare the results of CASSCF, SI-PDFT, CASPT2, MS-CASPT2, and XMS-CASPT2 calculations, we plotted their potential energy curves against each other in Figure 4. The energy for each method is plotted using the minimum energy on the ground-state curve as the zero of energy except for MS-CASPT2(ss), for which the ground-state energy of the ξ = -0.35 geometry (minimum-energy geometry of XMS-CASPT2) is used as the reference energy. Because CASPT2(sa), MS-CASPT2(sa), and XMS-CASPT2 relative energies differ by a maximum of only 9×10-4 eV for any geometry on the entire energy curves (Tables S8 and S13), among the three methods we only show the result of XMS-CASPT2 in Figure 4. CASPT2(ss) and MSCASPT2(ss) relative energies differ by a maximum of only 7×10-4 eV (Tables S10 and S15), so only MS-CASPT2(ss) is shown in Figure 4 to represent the “ss” method. As we can see from Figure 4, the “ss” method is problematic near the avoided crossing, but it is fixed by using the



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state-averaged density matrix to calculate H0. SI-PDFT closely resembles XMS-CASPT2 in the potential energy curve, while CASSCF differs more from these two methods. This demonstrates that SI-PDFT is comparable to CASPT2(sa), MS-CASPT2(sa), and XMS-CASPT2 in its ability to simulate mixed-valence compounds. In our studies, MS-CASPT2 and XMS-CASPT2 calculations for the (11,10) active space are too computationally expensive for the available contraction level in Molpro. For the (11,10) active space, using 2 processors it takes 87 hours and 87 GB memory to carry out a single-point MS-CASPT2 calculation for each geometry, while a single-processor calculation of SI-PDFT only takes 5-6 minutes and requires about 400 MB of memory. Therefore, we compare SI-PDFT from the (11,10) active space with MSCASPT2 and XMS-CASPT2 results from the (7,4) active space in Figure 4. The difference between CASPT2 relative energies for the (11,10) and (7,4) active spaces is only 6×10-3 eV or less for any geometry on the reaction path (Tables S8, S9, S13, S14), so this comparison is meaningful.


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Figure 4. Potential energy curves of the first two 2A2 states of the Spiro cation calculated by SA(2)-CASSCF(11,10) (blue dotted lines), SI-PDFT(11,10) (red dashed lines), MSCASPT2(ss)(7,4) (green dotted lines), and XMS-CASPT2(7,4) (black lines). The MSCASPT2(ss) and XMS-CASPT2 curves are almost on top of each other except for geometries near the avoided crossing, where MS-CASPT2(ss) fails. To further compare SI-PDFT and XMS-CASPT2 results with previously published NEVPT3 results, we summarize selective relative energies from these methods with the (7,4) and (11,10) active spaces in Table 1. The absolute energies and the relative energies of the entire potential energy curves are in Tables S6-S8 and Tables S11-S13, respectively. From the results



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of CASSCF, SI-PDFT, and NEVPT3, the (11,10) active space does not result in a qualitative difference in energy from the (7,4) active space, and the charge transfer behavior for the lowest two 2A2 states is already reasonably described by the (7,4) active space. However, the configurations in Table S3 show that the (11,10) active space is necessary when analyzing the characters of the wave function – results from the (11,10) active space clearly show that the electronic structure is more multiconfigurational for geometries more away from the D2d geometry, and becomes single-configurational for the D2d geometry. In the literature, the (7,4) active space is often used, but our results show that this active space is not adequate. Table 1 shows XMS-CASPT2 gives excitation energy at the saddle point similar to that for NEVPT3. As shown in Figure 4 and Table 1, although SI-PDFT slightly underestimates all the excitation energies on this approximated reaction path and the barrier height of the saddle point comparing to XMS-CASPT2 and NEVPT3, it is still closer to XMS-CASPT2 than CASSCF. In addition, as discussed above, SI-PDFT is much more affordable than MRPT methods when the active space is large. Therefore, we conclude that SI-PDFT is a promising alternative to MRPT methods, such as XMS-CASPT2 and NEVPT3, for multireference systems that have strongly interacting states, such as this difficult case of mixed-valence compound. It is even more advantageous than MRPT methods when the active space is large.

Table 1 Relative energy (in eV) for different methods. Energy of each state is relative to the “reference state”. ξmin represents the ξ value of the energy minimum of the ground state (12A2) for each particular method, which is ±0.45 for CASSCF, ±0.30 for SI-PDFT, and ±0.35 for XMSCASPT2. The 6-31G(d) basis set was used except for NEVPT3.


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Reference

Active Space

ξ

State

ξ

State

(7,4)

-0.5

22A2

-0.5

0

12A2

0

(11,10)

a

CASSCF

SI-PDFT

XMSCASPT2

NEVPT3(DZ)a

12A2

0.4965

0.3480

0.3972

0.4455

-0.5

12A2

0.0745

0.0019

0.0237

0.0864

22A2

-0.5

12A2

0.1717

0.0991

0.1302

0.1902

0

22A2

0

12A2

0.0972

0.0972

0.1064

0.1037

ξmin

22A2

ξmin

12A2

0.4489

0.2221

0.2882

-

0

12A2

ξmin

12A2

0.0745

0.0179

0.0314

-

0

22A2

ξmin

12A2

0.1717

0.1151

0.1378

-

-0.5

22A2

-0.5

12A2

0.4960

0.3392

-

0.4120

0

12A2

-0.5

12A2

0.0762

0.0053

-

0.0657

0

22A2

-0.5

12A2

0.1608

0.0898

-

0.1598

0

22A2

0

12A2

0.0846

0.0846

-

0.0941

ξmin

22A2

ξmin

12A2

0.4479

0.2139

-

-

0

12A2

ξmin

12A2

0.0768

0.0207

-

-

0

22A2

ξmin

12A2

0.1614

0.1052

-

-

NEVPT3 results are from Pastore et al.11 DZ represents the ANO-L basis set31 with 2s for H

and 3s2p for C and N. ξmin value for NEVPT3 is not reported, but can be read from the plot in Pastore et al.,11 which is between 0.35 and 0.40 and between -0.35 and -0.40. Conclusions Mixed-valence compounds have strong multireference character and their electronic states are strongly coupled. Previous studies show that the potential energy curves near the avoided crossing of the Spiro cation prototype mixed-valence compound can be reasonably described only by going to the third order NEVPT or by averaging the orbital energies between the two charge-localized electronic states in NEVPT2. In this study, we show that CASPT2, MSCASPT2, and XMS-CASPT2 can also obtain reasonable behavior of the energy curve by using



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the state-averaged density matrix in calculating the Fock operator of zeroth-order Hamiltonian. Then we show that SI-PDFT can correctly describe the potential energy curves near the avoided crossing of this prototype mixed-valence compound. These methods shown here to give good results provide new and general approaches in simulating the avoided crossings of mixed-valence compounds. In particular, SI-PDFT uses an on-top density functional to obtain the post-SCF correction and is much more computationally affordable than MRPT methods. Our results demonstrate that SI-PDFT is a promising method for simulating molecules that are large in size, have strongly interacting states, and undergo charge transfer processes.

g ASSOCIATED CONTENT SSupporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/ Additional computational details, Cartesian coordinates, CASSCF configurations and natural orbitals, absolute energies in hartrees, and relative energies in eV (PDF)


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g AUTHOR INFORMATION Corresponding Authors L.G., [email protected] D.G.T., [email protected] ORCID S. S. Dong 0000-0001-8182-6522 K. B. Huang 0000-0003-2003-725X L. Gagliardi 0000-0001-5227-1396 D. G. Truhlar 0000-0002-7742-7294 Author contributions † These authors contributed equally to this work Notes The authors declare no competing financial interest. g ACKNOWLEDGMENT KBH is supported by College of Chemistry, Nankai University. The work of all authors is supported by the Air Force Office of Scientific Research grant no. FA9550-16-1-0134.



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TOC Graphic

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The atomic numbering of the Spiro cation 171x120mm (72 x 72 DPI)



181x159mm (150 x 150 DPI)


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172x114mm (150 x 150 DPI)


The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51

22A2

12A2


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State-Interaction Pair-Density Functional Theory Can Accurately

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