Conical Intersection of the Ground and First Excited States of Water

ARTICLE pubs.acs.org/JPCA

Conical Intersection of the Ground and First Excited States of Water: Energies and Reduced Density Matrices from the Anti-Hermitian € dinger Equation Contracted Schro James W. Snyder, Jr. and David A. Mazziotti* Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, United States ABSTRACT: A conical intersection between the ground and first-excited states of water is computed through the direct calculation of two-electron reduced density matrices (2-RDMs) from solutions of the anti-Hermitian contracted Schr€ odinger equation (ACSE). This study is an extension of a previous study in which the ACSE was used to compute the energies around a conical intersection in the triplet excited states of methylene [Snyder, J. W., Jr.; Rothman, A. E.; Foley, J. J.; Mazziotti, D. A. J. Chem. Phys. 2010, 132, 154109]. We compute absolute energies of the 11A0 and 21A0 states of water (H2O) and the location of the conical intersection. The ACSE energies are compared to those from ab initio wave function methods. To treat multireference correlation, we seed the ACSE with an initial 2-RDM from a multiconfiguration self-consistent field (MCSCF) calculation. Unlike the situation for methylene, the two states in the vicinity of the conical intersection of water both have the same spatial symmetry. Hence, the study demonstrates the ability of the ACSE to resolve states of the same spatial symmetry that are nearly degenerate in energy. The 2-RDMs from the ACSE nearly satisfy necessary N-representability conditions. Comparison of the results from double-ζ and augmented double-ζ basis sets demonstrates the importance of augmented (or diffuse) functions for determining the location of the conical intersection.

I. INTRODUCTION A conical intersection is a point on a molecule’s potential energy surface in which two electronic states of the same spin multiplicity are energetically degenerate.1,2 This term is employed because the local topology near the intersection resembles a double cone or diabolo.2
4 Recent studies have demonstrated that low-lying conical intersections occur with far greater frequency than initially anticipated.2,4
22 This is particularly important because conical intersections enable the very efficient radiationless decay of a chemical system and play an important role in many photochemical and photobiological reactions.1,23 Conical intersections are known to exist in many complex, biological molecules, including luciferin in fireflies and the chromophore of green fluorescent protein, and are likely present on the potential energy surface of many other biologically important molecules.1,23
25 The conical intersection located between the 11A0 and 21A0 states of water (H2O) plays an important role in the homolytic dissociation of water into atomic hydrogen and the hydroxyl radical by accentuating nonadiabatic interactions.1,26
28 The treatment of this system is complicated because: (i) the ground and excited states have the same spatial symmetry and (ii) the excited state must be constrained to be orthogonal to the ground state. The former feature tends to complicate potential energy calculations in the vicinity of the conical intersection. The latter feature tends to increase the degree of multireference correlation29 along the 21A0 potential energy surface that contributes to the energy and its two-electron reduced density matrix (2-RDM). r 2011 American Chemical Society

In this paper we study the conical intersection along the 11A0 and 21A0 potential energy surfaces of water by solving the antiHermitian contracted Schr€odinger equation (ACSE) for 2-RDMs without explicitly computing many-electron wave functions.30
40 We determine both absolute energies of the 11A0 and 21A0 states of water (H2O) as well as the location of the conical intersection. Because the location of the conical intersection can be determined from its point-group symmetry,1 we do not implement more advanced methods for locating conical intersections based on derivative coupling or property matrices. This study augments a previous study in which the ACSE was used to analyze the energies around a conical intersection in the triplet excited states of methylene by demonstrating the ability of the ACSE to resolve states of the same spatial symmetry that are nearly degenerate in energy. Because the 2-RDM in the ACSE can be initialized from Hartree
Fock theory or a correlated method such as multiconfiguration self-consistent field (MCSCF),41 it can capture both single- and multireference correlation effects.32,35 For water, the ACSE yields energies that significantly improve upon those of the MCSCF and coupled cluster with single and double excitations (CCSD).42 Although the ACSE generally also improves upon multireference second-order perturbation theory (MRMP2),32,34,35,40,54,55 the ACSE in this case yields energies that are similar to those of both MRMP2 and completely Received: August 19, 2011 Revised: October 17, 2011 Published: November 11, 2011 14120

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renormalized coupled-cluster with perturbative triples excitations (CR-CC(T)). The ACSE is a 2-RDM-based method that directly computes a 2-RDM rather than an N-electron wave function. Importantly, the 2-RDM in the ACSE remains nearly N-representable throughout its solution; a 2-RDM is N-representable if and only if it derives from the integration of an N-electron density matrix. Other methods are also being actively developed to describe strong electron correlation.44
53 As in a previous theoretical study,55 calculations in augmented double-ζ basis sets show that the location of the conical intersection is sensitive to augmented (or diffuse) functions. The rest of this paper is organized as follows, In section II, we discuss the ACSE and its generalization to treat excited states. In section III, we present the data from the potential-energy surface scans conducted by various correlated methods, including the ACSE, and compare the accuracy of our results to the full configuration interaction (FCI) as a reference. In addition, we provide results from a larger augmented polarized valence double-ζ basis set in which the FCI cannot be computed.

II. THEORY After presenting the anti-Hermitian contracted Schr€odinger equation and the cumulant reconstruction of the three-electron RDM (3-RDM) in section IIA, we discuss the solution of the ACSE for ground- and excited-state energies and 2-RDMs in section IIB. A. ACSE. If the wave function of a given system is known, we can integrate the N-electron Schr€ odinger equation over all electrons save two to generate the contracted Schr€odinger equation (CSE),56
61 which can be expressed as ^ ¼ 2E 2D kl ÆΨj^a†i ^a†j ^al^ak HjΨæ ij

ð1Þ

^ is the Hamiltonian, Ψ is the N-electron wave function, where H each index denotes a spin orbital which is a product of a spatial orbital and a spin function, and ^a†i and ^ai are second quantized operators that create and annihilate an electron in the spin orbital i, respectively. The 2-RDM, 2Di,jk,l, can be expressed as 2

D k, l ¼ ÆΨj^a†i ^a†j ^al^ak jΨæ i, j

ð2Þ

The anti-Hermitian part of the CSE yields the ACSE,30
32,37,62 which only depends on the 2- and 3-RDMs. In a finite basis set, the ACSE can be formally expressed as ^ ÆΨj½^a†i ^a†j ^al^ak , HjΨæ ¼0

ð3Þ

To eliminate the 3-RDM from the ACSE, we can reconstruct the 3-RDM from the 2-RDM approximately56
58 according to its cumulant expansion63 3

i, j, k

j

i, j

j

D s, t, u ≈1 D is ∧ 1 D t ∧ 1 D ku þ 3ð2 D s, t
1 D is ∧ 1 D t Þ ∧ 1 D ku

ð4Þ

where the operator ∧ represents the antisymmetric tensor product known as the Grassmann wedge product.58 Substitution of the reconstructed 3-RDM into the ACSE produces an approximation to the ACSE that depends on only the 2-RDM and yet includes all second-order and many higher-order correlation terms.31 A further correction to the 3-RDM in terms of the 2-RDM37,57,63 is essential for highly accurate results in the ACSE’s single-reference formulation,30,31 but this correction is not needed in its multireference formulation. B. Solving the ACSE for Ground and Excited States. Consider a sequence of infinitesimal two-body unitary transformations

of an initial wave function |Ψ(λ)æ ^

jΨðλ þ εÞæ ¼ eεSðλÞ jΨðλÞæ

ð5Þ

where the transformations are ordered by a continuous time-like variable λ. For the transformation to be unitary the two-body operator ^S, defined by ^SðλÞ ¼

2 p, q Ss, t ðλÞa†p a†q at as ∑ p, q, s, t

ð6Þ

must be anti-Hermitian, ^S† =
^S. As ε f ∞, the changes in the energy and its 2-RDM with respect to λ are governed by the following differential equations:30
32 dE ^ ^SðλÞjΨðλÞæ ¼ ÆΨðλÞj½H, dλ

ð7Þ

and i, j

d2 D k, l ¼ ÆΨðλÞj½^a†i ^a†j ^al^ak , ^SðλÞjΨðλÞæ dλ

ð8Þ

To minimize the energy along λ, we select the elements of the two-particle matrix 2Sp,q s,t (λ) to minimize dE/dλ along its gradient with respect to these elements: 2 p, q S s, t ðλÞ

^ ¼ ÆΨðλÞj½^a†p ^a†q ^at ^as , HjΨðλÞæ

ð9Þ

Importantly, the left side of eq 9 is simply the residual of the ACSE. If the residual of the ACSE vanishes, the unitary transformations become identity operators, and the energy and 2-RDM cease to change with λ. Using the cumulant reconstruction of the 3-RDM in eq 463 permits us to express these equations approximately in terms of the 2-RDM. Hence, eqs 7
9 collectively provide a system of differential equations for evolving an initial 2-RDM to a final 2-RDM that solves the ACSE for stationary states.30
32 In practice, the equations are evolved in λ until either (i) the energy or (ii) the least-squares norm of the ACSE increases.32 As demonstrated in the recent extension of the ACSE to excited states,34,55 even though the unitary rotations are selected in eq 9 to minimize the energy, the system of differential equations in eqs 7
9 is capable of producing energy and 2-RDM solutions of the ACSE for both ground and excited states. Because excited states correspond to local energy minima of the ACSE34 and the gradient in eq 9 leads to a local rather than global energy minimum, an excited-state solution can be readily obtained from a good guess for the initial 2-RDM. A guess will be good when it is closer to the minimum of the desired solution of the ACSE than to any other minimum. Such 2-RDM guesses can be generated from multiconfiguration self-consistent-field (MCSCF) calculations. The initial MCSCF 2-RDM directs the optimization of the ACSE to a desired excited state because it contains important multireference correlation effects that identify the state. Importantly, even when the states are close or degenerate in energy, the ACSE can separate the two states as long as their 2-RDMs are distinguishable by at least one twobody operator, which practically is almost always the case. Additional details of the extension of the ACSE to excited states are given in ref 34. 14121

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III. APPLICATIONS After discussing additional computational details, we apply the ACSE to the calculation of the 11A0 and 21A0 potential energy surfaces of water. A. Computational Details. The potential energy surfaces of the 11A0 and 21A0 states of water were calculated near a conical intersection using the multireference solution to the ACSE and other computational methods. Results were compared with those from the following methods: multireference self-consistent field (MCSCF),41 multireference second-order many-body perturbation theory (MRMP2),54 coupled cluster with single and double excitations (CCSD),42 completely renormalized coupled-cluster with perturbative triples excitations (CR-CC(T)),42,43 and full configuration interaction (FCI). Calculations were performed in the correlation-consistent polarized valence double-ζ (ccpVDZ)64 basis set. Additional calculations were performed in the augmented correlation-consistent polarized double-ζ (augcc-pVDZ)65 basis set, in which the FCI could no longer be computed. The MCSCF method divides orbitals into three classes: (i) core orbitals that are fully occupied, (ii) active orbitals that are partially occupied, and (iii) virtual orbitals that are completely unoccupied. This division is done self-consistently

Figure 1. Schematic representation of water in Jacobi coordinates.

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to lower the energy of the molecule. All MCSCF calculations, including those used to seed the ACSE, employed a (6,8) active space as in previous work on the conical intersection of water,1 which means there are eight electrons in six active orbitals. 66 For the MCSCF to converge at all geometries, state averaging with a weight of [1,1] was used. There was essentially no difference in the energies calculated with and without state averaging for the few calculations that converged without state averaging. The experimental geometry was parametrized using a Jacobi coordinate scheme with the distance between the two hydrogen atoms fixed at RHH = 2.5832 Å. Jacobi coordinates, shown pictorially in Figure 1, are usually denoted by [R,r,γ], where r = RHH, while R and γ are the polar coordinates for the line connecting the carbon atom to the center of mass of the two hydrogen atoms.2 The geometry at the conical intersection is slightly off-linear (γ = 0.00021).1 The parameter R was increased incrementally from the linear geometry at intervals of 0.05 Å. The ACSE calculations were computed using a developmental code, while all other calculations were performed using the GAMESS electronic structure package.66 B. Results. 1. Energies Near the Conical Intersection in the ccpVDZ Basis Set. The ACSE was applied to compute the potential energy surfaces of the ground and first-excited states of water, which are 11A0 and 21A0 , respectively, near the conical intersection. In Figure 2 and Table 1, the differences between the potential-energy surfaces of the MCSCF, MRMP2, CCSD, CR-CC(T), and ACSE and the FCI are displayed. For the 11A0 state, the MCSCF, MRMP2, CCSD, CR-CC(T), and ACSE methods have maximum energy deviations of 184.08, 0.93, 6.59, 1.58, and 3.63 milliHartrees (mH), respectively, and minimum energy deviations of 179.47, 0.81, 5.36, 1.10, and 0.08 mH.

Figure 2. MCSCF (a), MRMP2 (b), CR-CC(T) (c), and ACSE (d) potential energy curves for the 11A0 and 21A0 states of water, as functions of R, plotted against those from FCI, given by data points. 14122

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Table 1. Deviations in Energy (mH) of MCSCF, MRMP2, CR-CCSD, CR-CC(T), and the ACSE from FCI are Reported for the 11A0 and 21A0 States of Water (H2O) in the cc-pVDZ Basis Set

state (Å) 1 0

1A

0.0

21A0

11A0

FCI energy (H)

lowest eigenvalues state

deviation from FCI (mH) R

Table 3. Lowest 2D, 2Q, and 2G Eigenvalues are Reported for the 11A0 and 21A0 States of Water

MCSCF MRMP2 CCSD CR-CC(T) ACSE


76.0426

184.08

0.93

5.36

1.10

3.63

R (Å)

2

D

2

Q

2

G

0.0


1.3e
3


5.7e
5


7.0e
4

0.1


1.4e
3


6.7e
5


8.7e
4

0.2 0.3


1.7e
3
4.1e
3


9.8e
5
1.6e
4


1.2e
3
2.3e
3

0.1


76.0414

183.43

0.90

5.41

1.13

3.25

0.4


76.0368


4.0e
3


1.5e
4


2.1e
3

0.2

180.83

0.93

5.56

1.21

2.73

0.5


3.8e
3


1.4e
4


1.9e
3

0.3


76.0246

181.67

0.91

5.81

1.32

1.41

0.0


75.9964


2.0e
4


2.8e
3

180.63

0.87

6.15

1.44


0.08


4.5e
3

0.4

0.1


2.8e
3

179.47 163.97

0.81
0.08

6.59 10.16

1.58 3.74


1.50
2.52


1.8e
4

0.5 0.0


75.9360
75.9880


4.4e
3

0.2


4.3e
3


1.7e
4


2.6e
4

0.3


2.2e
3


1.7e
4


1.6e
3

0.4 0.5


2.8e
3
3.3e
3


3.3e
4
4.9e
4


2.2e
3
2.6e
3

0.1


75.9966

162.05

0.52

11.02

3.77


2.12

0.2


76.0145

160.49

1.28

12.60

3.82


2.36

0.3


76.0288

156.24

1.79

13.96

3.87


2.80

0.4


76.0270

154.44

2.20

14.96

3.95


3.13

0.5


75.9914

152.85

2.46

15.70

4.05


3.36

Table 2. Natural Occupation Numbers are Reported for the 11A0 and 21A0 States of Water in the cc-pVDZ Basis Set from the Multiconfiguration Self-Consistent Field (MCSCF), Coupled Cluster with Singles Doubles Excitations (CCSD), Anti-Hermitian Contracted Schr€ odinger Equation (ACSE), and Full Configuration Interaction (FCI) Methods occupation numbers of natural orbitals MCSCF orbital index

0

1A

2A

CCSD 0

1A

0

ACSE 0

2A

0

1A

FCI 0

2A

0

1A

2A0

1

1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 1.0000 1.0000

2

0.9993 0.9998 0.9895 0.9967 0.9882 0.9912 0.9888 0.9912

3

0.9993 0.9995 0.9583 0.9923 0.9855 0.9870 0.9845 0.9870

4

0.9853 0.9872 0.9856 0.9977 0.9855 0.9775 0.9845 0.9791

5

0.9413 0.5000 0.9856 0.5003 0.9322 0.4993 0.9511 0.4993

6

0.0572 0.4999 0.0377 0.4997 0.0633 0.4967 0.0447 0.4969

7

0.0077 0.0136 0.0106 0.0054 0.0122 0.0187 0.0115 0.0166

8

0.0000 0.0000 0.0090 0.0015 0.0089 0.0086 0.0096 0.0084

9

0.0000 0.0000 0.0090 0.0015 0.0084 0.0042 0.0096 0.0041

10

0.0000 0.0000 0.0029 0.0012 0.0029 0.0035 0.0032 0.0037

For the 21A0 state, the MCSCF, MRMP2, CCSD, CR-CC(T), and ACSE methods have maximum energy deviations of 163.97, 2.46, 15.70, 4.05, and 3.36 mH, respectively, and minimum energy deviations of 152.85, 0.08, 10.16, 3.74, and 2.12 mH. In relation to the FCI, the ACSE energies significantly improve upon the accuracy of the MCSCF and CCSD energies. The natural occupation numbers from the ACSE are compared in Table 2 with those from MCSCF, CCSD, and FCI; for both ground and excited states the ACSE like FCI captures more electron correlation than CCSD. Importantly, the occupation numbers reflect that the energies of the states 11A0 and 21A0 have a crossing rather than an avoided crossing. Before the conical intersection the excited state is a biradical while after the conical intersection the ground state is a biradical. For

21A0

consistency, we label this biradical state 21A0 both before and after the intersection. The MRMP2, CR-CC(T), and ACSE energies for the 11A0 and 21A0 potential-energy surfaces of water are quite similar in accuracy. Typically, the ACSE is more accurate than MRMP2 because the ACSE with cumulant 3-RDM reconstruction incorporates correlation effects from many higher orders of a renormalized perturbation theory. While this improvement was noted in previous ACSE studies (i.e., see refs 32, 34, 35, 40, and 55), it is not present here because the active space is very large in relation to the molecule’s size, making MRMP2 highly accurate. Calculations in smaller active spaces are not reported for the potential energy surface due to MCSCF convergence issues at selected geometries, but for the geometries that did converge, the energies from the ACSE are more accurate than those from MRMP2. The accuracy of the shape of a potential energy curve can be judged approximately by its nonparallelity error (NPE), which is the difference between the highest and lowest energy deviations. The MCSCF, MRMP2, CCSD, CR-CC(T), and ACSE methods have NPEs of 4.61, 0.12, 1.23, 0.48, and 3.28 mH for the 11A0 and 11.12, 2.38, 5.54, 0.31, and 1.24 mH for the 21A0 . Although the MRCI was not applied here, it has recently been shown to compute energies that are comparable to the ACSE for small molecules but less accurate than the ACSE for larger molecules. Because the MRCI scales exponentially with respect to the number of active orbitals, it is generally more expensive than the ACSE.32 The 11A0 and 21A0 state 2-RDMs produced by the ACSE maintain the N-representability of the 2-RDM within the accuracy of the 3-RDM reconstruction. A significant set of necessary N-representability constraints, known as 2-positivity conditions,53,67
72 requires keeping the eigenvalues of three different forms of the 2-RDM, the 2D, 2Q, and 2G matrices, nonnegative. These three sets of eigenvalues correspond to probability distributions for two particles, two holes, as well as one particle and one hole, respectively. Table 2 provides the lowest eigenvalues of these matrices for the ground and first-excited states of water at the conical intersection, normalized to N(N
1), (r
N)(r
N
1), and N(r
N + 1), respectively, where r is the rank of the spin
orbital basis set. The most negative eigenvalues are 2.5
4 orders of magnitude smaller than the largest positive eigenvalues, which are on the order of unity save for the largest eigenvalue of 2G that is on the order of N. 14123

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Figure 3. MRMP2 (a) and CR-CC(T) (b) potential energy curves for the 11A0 and 21A0 states of water, as functions of R, plotted against the ACSE.

Table 4. Energies Relative to the MCSCF (mH) are Reported for the 11A0 and 21A0 States of Water in the aug-cc-pVDZ Basis Set Ex
EMCSCF (mH) state 1 0

1A

21A0

R (Å)

MCSCF energy (H)

MRMP2

CR-CC(T)

ACSE

0.0


75.8745


210.31


216.25


202.41

0.1


75.8741


209.86


215.53


202.42

0.2


75.8708


207.49


214.22


202.72

0.3


75.8592


206.83


213.49


203.07

0.4 0.5


75.8324
75.7735


205.47
203.88


212.16
210.67


203.36
202.51

0.0


75.8308


196.05


176.07


184.17

0.1


75.8412


186.25


173.70


181.46

0.2


75.8621


178.77


169.87


177.87

0.3


75.8795


172.61


166.11


174.47

0.4


75.8799


168.53


163.49


172.05

0.5


75.8466


165.03


160.75


169.67

2. Energies near the Conical Intersection in the aug-cc-pVDZ Basis Set. In this section, we report the results from applying the ACSE to the 11A0 and 21A0 states of water in the aug-cc-pVDZ65 basis set. Due to the larger basis set size, we are no longer able to compute the FCI energies. As such, the ACSE energies are compared to the MCSCF, MRMP2, and CR-CC(T) energies, which were calculated using GAMESS. The potential energy curves from the MRMP2, CR-CC(T), and ACSE methods are compared in Figure 3 and Table 4. In

Figure 4. MCSCF (top) and ACSE (bottom) three-dimensional potential energy curves for the 11A0 and 21A0 states of water as functions of R and γ (deg).

addition, a three-dimensional plot of the MCSCF and ACSE potential energy surfaces is presented in Figure 4. Both of the MCSCF potential energy surfaces are higher in energy than the ACSE potential energy surfaces. The CR-CC(T) ground-state potential energy surface is lower in energy than the ACSE potential energy surface, while the CR-CC(T) excited-state potential energy surface is higher than the corresponding ACSE surface. Neither of the MRMP2 potential energy surfaces is consistently higher or lower than the ACSE potential energy surface. The ACSE potential energy surface tends to be lower than the MRMP2 potential energy surface at geometries following the conical intersection. In the aug-cc-pVDZ basis set, the conical intersection from the ACSE is located at approximately [0.33, 2.5832, 0.00021], which is in good agreement with the location [0.35, 2.5832, 0.00021] 14124

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The Journal of Physical Chemistry A published by Yarkony.1 The conical intersections from the MCSCF, MRMP2, and CR-CC(T) methods are located at approximately [0.27, 2.5832, 0.00021], [0.34, 2.5832, 0.00021], and [0.40, 2.5832, 0.00021]. These results are consistent with our previous study of methylene,55 which suggested that augmented or diffuse basis functions perform a critical role in stabilizing the heavier atom at larger distances from the hydrogen atoms center of mass.

IV. DISCUSSION AND CONCLUSIONS The conical intersection between the 11A0 and 21A0 states of water has been studied by computing 2-RDMs from the ACSE without constructing many-electron wave functions. The calculations utilize extensions of the ACSE to excited states.34 We computed the potential energy curves of the 11A0 and 21A0 states of water including the location of the conical intersection (Table 3). The location of the intersection computed by the ACSE in the aug-cc-pVDZ basis set65 corroborates Yarkony’s results with a high degree of accuracy. Even though the ground and excited state of water exhibit the same spatial symmetry, the ACSE was able to distinguish between the two states in the vicinity of the conical intersection where they are nearly degenerate in energy. In accordance with previous works, the ACSE improves the accuracy of MCSCF by 2 orders of magnitude and CCSD by 1 order of magnitude.35,55 The ACSE is comparable to CR-CC(T), with slightly higher and lower energies calculated on the groundand first-excited potential energy surface.40 In contrast to previous works, the ACSE method is comparable to the MRMP2 method.35,40,55 This is attributable to the size of the active space. The average differences in the MCSCF, MRMP2, CCSD, CRCC(T), and ACSE ground-state energies from FCI are 181.69, 0.89, 5.81, 1.30, and 1.78 mH, respectively. The differences in the excited-state energies are 158.34, 1.36, 13.07, 3.87, and
2.97 mH, respectively. Similar accuracy was obtained at the conical intersection. Furthermore, the ACSE produced 2-RDMs that are almost N-representable, which is evident from the slight deviations from the well-known 2-positivity conditions. Both single- and multireference correlation can be accurately captured with the ACSE, which is especially important for excited states where multiple determinants can contribute substantially to the wave function at zeroth order. In the multireference formulation of the ACSE, employed here, the ACSE is seeded with an initial MCSCF 2-RDM in which the orbitals are either active (correlated) or inactive (uncorrelated). The ACSE correlates the active orbitals with the inactive orbitals. Because the ACSE contains many high orders of a renormalized perturbation theory, it yields energies that are more accurate than MCSCF and often MRMP2. Furthermore, even when the electronic system lacks significant multireference correlation, the ACSE with a MCSCF 2-RDM guess generally yields energies that lie in accuracy between those from coupled cluster with single
double excitations (CCSD) and those from CCSD plus triple excitations (CCSDT). Computationally, the ACSE scales in floating-point operations as (rc + ra)2r4v , where rc, ra, and rv are the numbers of core, active, and virtual orbitals, respectively. In contrast, traditional multireference methods like MCSCF, MRPT2, and MRCI depend on the number of determinants in the active space and, hence, scale exponentially with the number of orbitals in the active space. Although the ACSE was originally restricted to singlet, ground-state

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calculations, its recent extension to excited states34 and arbitrary spin states35 opens further possibilities for the accurate study of conical intersections and other important excited-state phenomena.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT J.W.S expresses his appreciation to James and Irene Snyder for their support and encouragement. J.W.S thanks the Barry Goldwater Scholarship and Excellence in Education Foundation for their support. D.A.M gratefully acknowledges support from the National Science Foundation (NSF), the Army Research Office (ARO), the David and Lucile Packard Foundation, the HenryCamille Dreyfus Foundation, and Microsoft Corporation. ’ REFERENCES (1) Yarkony, D. R. Mol. Phys. 1998, 93, 971–983. (2) Yarkony, D. R. Rev. Mod. Phys. 1984, 68, 985–1013. (3) Berry, M. V.; Wilkinson, M. Proc. R. Soc. A 1984, 392, 15. (4) Bernardi, F.; Olivucci, M.; Robb, M. A. J. Chem. Soc. Rev. 1996, 25, 321–328. (5) Bearpark, M. J.; Robb, M. A.; Schlegel, H. B. Chem. Phys. Lett. 1994, 223, 269–274. (6) Bearpark, M. J.; Bernardi, F.; Clifford, S.; Olivucci, M.; Robb, M. A.; Smith, B. R.; Vreven, T. J. Am. Chem. Soc. 1996, 118, 169. (7) Ostojic, B.; Domcke, W. Chem. Phys. 2001, 269, 1. (8) Krawczyk, R. P.; Malsch, K.; Hohlneicher, G.; Gillen, R. C.; Domcke, W. Chem. Phys. Lett. 2000, 320, 535. (9) Boggio-Pasqua, M.; Bearpark, M. J.; Klene, M.; Robb, M. A. J. Chem. Phys. 2004, 120, 7849. (10) Levine, B. G.; Coe, J. D.; Martinez, T. J. J. Phys. Chem. B 2008, 112, 405. (11) Levine, B. G.; Martinez, T. J. J. Phys. Chem. A 2009, 113, 12815. (12) Sakai, S.; Yamada, T. Phys. Chem. Chem. Phys. 2008, 10, 3861. (13) Norton, J. E.; Houk, K. N. Mol. Phys. 2006, 104, 993. (14) Bachler, V. J. Comput. Chem. 2004, 25, 343. (15) Fuss, W.; Schmidt, W. E.; Trushin, S. A. Chem. Phys. Lett. 2001, 342, 91. (16) Sakai, S. Chem. Phys. Lett. 2000, 319, 687. (17) Izzo, R.; Klessinger, M. J. Comput. Chem. 2000, 21, 52. (18) Zilberg, S.; Haas, Y. J. Phys. Chem. A 1999, 103, 249. (19) Sakai, S. Chem. Phys. Lett. 1998, 287, 263. (20) Matsika, S.; Yarkony, D. R. J. Phys. Chem. A 2002, 106, 2580. (21) Dallos, M.; Lischka, H.; Shepard, R.; Yarkony, D. R.; Szalay, P. G. J. Chem. Phys. 2004, 120, 7330. (22) Matsika, S.; Yarkony, D. R. J. Am. Chem. Soc. 2003, 125, 10672. (23) Olivucci, M.; Ragazos, I. N.; Bernardi, F.; Robb, M. A. J. Am. Chem. Soc. 1993, 115, 3710–3721. (24) Burr, J. G. Chemi- and Bioluminescence; Marcel-Dekker: New York, 1985. (25) Toniolo, A.; Olsen, S.; Manohar, L.; Martínez, T. J. Faraday Discuss. 2004, 127, 149–163. (26) Dobbyn, A. J.; Knowles, P. J. Mol. Phys. 1997, 91, 6.  ; Halasz, G. J.; Baer, M. Chem. Phys. Lett. 2004, (27) Vibok, A 399, 7–14. (28) Kaduk, B.; Van Voorhis, T. J. Chem. Phys. 2010, 133, 061102. (29) Jensen, F. Introduction to Computational Chemistry, 2nd ed.; Wiley: New York, 2007. (30) Reduced-Density-Matrix Mechanics with Application to ManyElectron Atoms and Molecules. In Advances in Chemical Physics; Mazziotti, D. A., Ed.; Wiley: New York, 2007; Vol. 134. 14125

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(72) Nakata, M.; Braams, B. J.; Fujisawa, K.; Fukuda, M.; Percus, J. K.; Yamashita, M.; Zhao, Z. J. Chem. Phys. 2008, 128, 164113.

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