Multiconfiguration Pair-Density Functional Theory ... - ACS Publications

Letter pubs.acs.org/JPCL

Multiconfiguration Pair-Density Functional Theory Is as Accurate as CASPT2 for Electronic Excitation Chad E. Hoyer, Soumen Ghosh, Donald G. Truhlar,* and Laura Gagliardi* Department of Chemistry, Chemical Theory Center, and Minnesota Supercomputing Institute, University of Minnesota, 207 Pleasant Street SE, Minneapolis, Minnesota 55455-0431, United States S Supporting Information *

ABSTRACT: A correct description of electronically excited states is critical to the interpretation of visible−ultraviolet spectra, photochemical reactions, and excited-state charge-transfer processes in chemical systems. We have recently proposed a theory called multiconfiguration pair-density functional theory (MC-PDFT), which is based on a combination of multiconfiguration wave function theory and a new kind of density functional called an on-top density functional. Here, we show that MC-PDFT with a firstgeneration on-top density functional performs as well as CASPT2 for an organic chemistry database including valence, Rydberg, and charge-transfer excitations. The results are very encouraging for practical applications.

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affordable choices of configurations include only a small portion of the dynamic electron correlation, they do not yield chemical accuracy for excitation energies, but they can serve as wellbalanced reference functions for post-SCF treatments of the dynamic correlation energy. In wave function theory, this is usually done by perturbation theory (CASPT2)17 or multireference configuration interaction with single and double excitations (MRCISD). In MC-PDFT it is done by using an ontop density functional that depends on total density and the ontop pair density. This is more economical than CASPT2 or MRCISD. One should, however, bear in mind that MC-PDFT inherits all the difficulties of multiconfiguration SCF, which often requires system-dependent choices for the number of electrons in the active space, the number of orbitals in the active space, and the number of states averaged, as well as weights used in the average. The main advantage over multiconfiguration wave function theory is that MC-PDFT allows one to replace the post-SCF second-order perturbation theory step or multireference configuration interaction step by a less expensive density functional energy calculation. Preliminary tests of MC-PDFT for electronic excitation energies have been very promising.14,18−20 Here, we present a systematic test of MC-PDFT for electronic excitations of the most important kinds of chromophores in organic chemistry. Specifically, we test using CASSCF wave functions with the ontop density functionals tPBE,14 ftPBE,20 trevPBE, and ftrevPBE, where tPBE is a translated form of the PBE21 exchangecorrelation functional; ftPBE is a fully translated form of PBE;

omputational modeling of electronically excited states is important for studying photochemistry,1 photocatalysis,2 biological systems,3 and organic electronics.4,5 Kohn−Sham density functional theory (KS-DFT) has become the popular choice for studying ground-state properties of large molecular systems.6 Although linear-response time-dependent Kohn− Sham density functional theory (TD-KS-DFT)7 extends DFT to electronically excited states, practical calculations are almost always based on the adiabatic approximation, which means that frequency-independent ground state exchange-correlation functionals are used in the time-dependent problem. With this approximation, accurate excitation energies can be obtained for many excitations to valence-excited states. However, chargetransfer and Rydberg excitations tend to be poorly described.8−13 These failures of TD-KS-DFT are often attributed to the wrong long-range behavior of exchangecorrelation functionals. Many modern functionals with longrange Hartree−Fock exchange provide better accuracy for these challenging excited states, but, though they are capable in principle of describing multiconfigurational wave functions of various types, in practice, they do not always provide satisfactory performance with available exchange-correlation functionals and the linear-response approximation. This is perhaps due to the difficulty of representing inherently multiconfigurational excited states in terms of excitations from one Slater determinant. An alternative to TD-KS-DFT is multiconfigurational pairdensity functional theory (MC-PDFT).14 This is a timeindependent method based on multiconfiguration self-consistent field (MCSCF)15 wave functions, such as the complete active space SCF (CASSCF)16 method, which properly describes excited states as multiconfigurational with the correct spin and spatial symmetry. Since MCSCF wave functions with © XXXX American Chemical Society

Received: December 14, 2015 Accepted: January 21, 2016

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DOI: 10.1021/acs.jpclett.5b02773 J. Phys. Chem. Lett. 2016, 7, 586−591

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The Journal of Physical Chemistry Letters Table 1. Excitation Energies, in eV, with Moderate Active Spacesa acetaldehyde acetone formaldehyde pyrazine pyridazine pyridine pyrimidine s-tetrazine ethylene butadiene benzene napthalene furan hexatriene water pNA DMABN B-TCNE MSEe valence MUEf valence MSEe Rydberg MUEf Rydberg MSEe CT MUEf CT Averageh MSE Averageh MUE

excited state

active spaceb

CASSCF

CASPT2

tPBEc

PBEd

PBE0d

reference

A″ n → π* 1 A2 n→ π* 1 A2 n→ π* 1 B3u n→ π* 1 B1 n→ π* 1 B1 n→ π* 1 B1 n→ π* 1 B3u n→ π* 1 B1u π→ π* 1 Bu π→ π* 1 B2u, π→ π* 3 B1u, π→ π* 1 B3u, π→ π* 3 B2u, π→ π* 1 B2, π→ π* 3 B2, π→ π* 1 Bu, π→ π* 3 Bu, π→ π* Singlet, 2px→ 3s Triplet, 2px→ 3s Intramolecular CT,1A1, π→ π* Intramolecular CT,1A1, π→ π* Intermolecular CT,1A, π→ π*

SA(2)-(12,12) SA(2)-(12,12) SA(2)-(12,10) SA(3)-(10,10) SA(2)-(10,10) SA(3)-(8,8) SA(2)-(10,10) SA(2)-(14,14) SA(5)-(4,10) SA(4)-(10,15) SA(2)-(6,13) (6,13) SA(2)-(10,10) (10,10) SA(2)-(6,10) (6,10) SA(2)-(6,12) (6,12) SA(2)-(8,9) (8,9) SA(3)-(12,12) SA(3)-(12,12) SA(2)-(4,4)

4.56 4.71 4.75 5.08 4.63 5.64 4.97 3.65 7.72 6.38 4.99 3.94 4.24 3.04 6.68 3.93 5.78 2.88 7.69 7.16 6.09 6.18 4.48 0.44 0.53 0.23 0.23 1.43 1.43 0.70 0.73

4.27 4.44 3.92 4.04 3.67 5.06 4.38 2.56 8.16 6.51 4.83 3.98 4.21 3.18 6.32 4.08 5.32 2.67 7.61 7.19 4.62 4.87 3.84 0.11 0.15 0.20 0.20 0.29 0.29 0.20 0.21

3.92 4.35 3.55 3.86 3.24 4.80 4.06 2.48 6.77 6.03 5.09 4.27 4.40 3.31 6.51 4.20 5.48 2.74 7.46 7.09 3.92 4.28 3.63 −0.03 0.29 0.07 0.07 −0.21 0.24 −0.06 0.20

4.10 4.20 3.77 3.52 3.11 4.32 3.75 1.84 7.35 5.41 5.14 3.91 4.02 2.79 5.87 3.88 4.42 2.27 6.36 6.01 3.55 4.36 1.35 −0.33 0.36 −1.02 1.02 −1.07 1.07 −0.81 0.82

4.25 4.40 3.90 3.96 3.63 4.83 4.31 2.29 7.46 5.63 5.37 3.49 4.38 2.54 6.04 3.57 4.67 1.91 7.13 6.69 4.16 4.77 2.08 −0.17 0.29 −0.29 0.29 −0.48 0.62 −0.31 0.40

4.2832 4.4332 4.0032 3.9733,34 3.6032 4.7435,36 4.1837 2.2532 8.0238 6.2139 4.9040 4.1241 4.0042 3.1141 6.0643 4.1741 4.9344 2.6941 7.4045 7.0046 4.30g 4.5747 3.5948

1

a

See Supporting Information for details of active space choices. b(n,m) specifies the active space, where n is the number of electrons, m is the number of orbitals, and SA(N) indicates that the state average is over N states, where SA stands for state averaging. cMC-PDFT. dLinear-response timedependent Kohn−Sham density functional theory is used. eMean signed error. fMean unsigned error. gδ-CR-EOMCC(2,3),D49,50/6-31+G**. 1 1 h1 valence + 3 Rydberg + 3 charge transfer 3

and (f)trevPBE is a (fully) translated form of the revPBE22 exchange-correlation functional. These functionals are firstgeneration on-top density functionals, so this study provides a baseline understanding of the performance of the theory for spectroscopy. The tPBE on-top functional tested in the article proper derives from the PBE21 exchange-correlation functional, which has no empirical parameters, and there are no parameters in the translation; the physical basis of the translation is the relationship between the on-top pair density and the spin densities for a broken-symmetry Slater determinant, as described by Perdew et al.23 Although the functional can be improved in future work by making it more flexible and optimizing parameters against theoretical considerations, experimental data, or both, here, we test the functional in its simplest form based solely on translation of an existing functional by a previously published protocol. The purpose of this paper is therefore to provide mainly a test of the physicality of the MC-PDFT method rather than a test of parametrization. The Supporting Information presents results obtained with other functionals, in particular trevPBE, ftPBE, and ftrevPBE. The trevPBE on-top functional is obtained from the revPBE22 exchange-correlation functional by the same prescription as

used to obtain tPBE from PBE, and revPBE was obtained22 from PBE by changing one parameter to improve the agreement of exchange-only total atomic energies with exact exchange-only results with the motivation of improving both atomic total energies and molecular atomization energies. The ft translation protocol used to obtain the ftPBE and ftrevPBE on-top functionals corresponds to a more advanced translation prescription, to include the gradient of the on-top density, and it has five parameters adjusted primarily to obtain a smooth functional that was tested for good performance for 61 groundstate properties and one electronic excitation energy.20 MC-PDFT is tested here for 23 excitations of 18 diverse molecules. We include the lowest-energy spin-conserving valence excitation energies of ten organic molecules (acetaldehyde, acetone, formaldehyde, pyrazine, pyridazine, pyridine, pyrimidine, s-tetrazine, ethylene, and butadiene).24 To these we added the lowest singlet and triplet valence excitation energies of benzene, naphthalene, furan, and all-E-hexatriene. Altogether, this makes 18 valence excitations for 14 different molecules. Our database also includes two Rydberg states, namely, the lowest singlet and triplet excitations of water. As discussed by Song and Hirao,13 intramolecular and intermolecular charge transfer present different kinds of challenges for 587


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The Journal of Physical Chemistry Letters Table 2. Large and Small Active Spaces, ASs, and State Averaging Specification state label

large AS

active orbitals in large AS

small AS

A″ n → π* 1 A2 n → π* 1 A2 n→ π* 1 B3u n → π* 1 B1 n → π* 1 B1 n → π* 1 B1 n → π* 1 B3u n → π* 1 B1u π → π* 1 Bu π → π* 1 B2u, π → π* 3 B1u, π → π* 1 B3u, π → π* 3 B2u, π → π* 1 B2, π → π* 3 B2, π → π* 1 Bu, π → π* 3 Bu, π → π* Singlet, 2px→ 3s Triplet, 2px→ 3s CT,1A1, π→ π* CT,1A1, π→ π* CT,1A, π→ π*


π,π*, 2(n,n*), σC−C, σC−C*, σC−O, σC−O*, σC−H, σC−H* π,π*, 2(n,n*), 2(σC−C,σC−C*), σC−O, σC−O* full valence 3(π,π*), 2(n,n*) 3(π,π*), 2(n,n*) 3(π,π*), n, n* 3(π,π*), 2(n,n*) 3(π,π*), 4(n,n*) π,π*, σC−C, σC−C*, 4(Ryd), 2 correlating π* π, π*, π, π*, 3(σC−C,σC−C*), Ryd, 4 correlating π* 3(π,π*), 7π*


1

acetaldehyde acetone formaldehyde pyrazine pyridazine pyridine pyrimidine s-tetrazine ethylene butadiene benzene napthalene furan hexatriene water pNA DMABN B-TCNE

5(π,π*) 2(π,π*), (n,n*), 4 π* 3(π,π*), 6 π* 2(σO−H, σO−H*), 2(n,n*), 3s 5(π,π*), (n,n*) 5(π,π*), (n,n*) 2(π,π*)

correct spatial or spin symmetry. MC-PDFT represents the densities and on-top pair densities of all states by multiconfigurational wave functions of the correct spatial and spin symmetry. However, MC-PDFT has the added cost of configuration interaction (CI) coefficient optimization during the MCSCF calculation, which dominates the cost in the limit of large configuration expansions. All the systems investigated have closed-shell ground states. We used CASSCF as the MCSCF method. The details of the calculations are reported in Supporting Information. Although CASSCF is not expected to yield accurate energetics, it is included in the comparisons since CASPT2 and tPBE both use the CASSCF reference wave function. Because the PBE exchange-correlation functional is used in tPBE, we also present results obtained by TD-KS-DFT using PBE and PBE0, where PBE025 is obtained by replacing 25% of PBE’s local exchange with Hartree−Fock exchange. For each molecule, we used an active space that includes the main configurations needed to describe the excitations; these results are reported in Table 1 and the active spaces are specified in Table 2, where the notation SA(k)-(m,n) denotes state averaging over k states with m electrons in n active orbitals. As an exploratory exercise, we also considered very small active spaces, and those results are reported in Supporting Information. First, we consider the results for 18 valence excitations in Table 1. As expected, CASSCF performs poorly for these excitations with a 0.53 eV mean unsigned error (MUE). The tPBE functional gives an MUE of only 0.29 eV and does as well as PBE0 and better than PBE. Note that tPBE and PBE are local functionals whereas PBE0 has nonlocal exchange. CASPT2 performs the best among all the methods if consideration is limited to valence excitations. Next, we consider the results in Table 1 for the lowest singlet and triplet excitations of water, which are Rydberg in nature.

density functional theory, and our test set includes both kinds. In particular, we include p-nitroaniline (pNA) and 4(dimethylamino)benzonitrile (DMABN), which have intramolecular charge transfer excitations, and the donor−acceptor complex of benzene (B) and tetracyanoethylene (TCNE), which has an intermolecular charge transfer excitation. This extensive data set allows us to test performance of MC-PDFT for all the main types of excitations of main-group molecules. The tests in this paper are only for excitations from valence orbitals. We have not yet performed any calculations involving excitations from core orbitals. We briefly review MC-PDFT here, and a more detailed description can be found in ref 14. The kinetic energy, density, and on-top pair-density are taken from a converged MCSCF calculation, and the other electronic energy contributions are computed post-SCF as functionals of the density and on-top pair-density E[ρ , Π] =

∑ hpqDpq + ∑ gpqrsDpqDrs + Eot[ρ , Π] + VNN pq

pqrs

(1)

where ρ is the density; Π is the on-top pair density; p, q, r, and s are generic orbital indices; hpq are the one-electron integrals; Dpq is a one-electron density matrix element; gpqrs are the twoelectron integrals; Eot is the on-top energy; and VNN is the internuclear repulsion. The on-top energy is analogous to the exchange-correlation energy in Kohn−Sham density functional theory (KS-DFT), and it includes a correction to the MCSCF kinetic energy plus exchange and electron correlation. An important aspect of the comparison of MC-PDFT to TDKS-DFT is their treatment of symmetry. The TD-KS-DFT method represents the electron density of the ground state as a single Slater determinant and the excited states as the linear response of that determinant to a frequency-dependent field, and neither ground nor excited states necessarily have the 588


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The Journal of Physical Chemistry Letters (The reason why only Rydberg excited states are present26 may be understood by comparison to the united-atom limit of Ne, which can only have excitations in which a principal quantum number increases.) Local KS-DFT functionals underestimate Rydberg excitations because of the wrong asymptotic behavior of the exchange−correlation potential, and consequently, PBE performs poorly for these two excitations. Because of the 25% Hartree−Fock exchange, PBE0 performs relatively better. But, tPBE performs remarkably well (MUE of only 0.07 eV), which is even better than CASPT2 when the same large active spaces specified in Table 2 are used. In previous work,19 we have also shown that MC-PDFT is stable to adding diffuse functions, which is important when one treats Rydberg states. Charge-transfer (CT) excitations represent a great challenge for KS-DFT8−13 (see Supporting Information for a discussion). Previously, we have shown that MC-PDFT performs well for long-range intermolecular charge transfer excitations that are in the small-spatial-overlap region. As reported in Table 1, we have tested tPBE and other methods for both intramolecular and intermolecular excitations. The tPBE functional performs very well, with an MUE of 0.24 eV. For the intermolecular excitation, the spatial overlap is very small, but the spatial overlap is much higher for the intramolecular excitations. The PBE and PBE0 methods perform relatively well for the largespatial-overlap excitations, but they fail completely for the small-spatial-overlap excitations. The tPBE method, on the other hand, performs well for all extents of spatial overlap, indicating that the performance of the on-top density functional for different excitations does not depend strongly on the extent of spatial overlap between occupied and virtual orbitals. In practical applications, general spectra contain a mix of valence, Rydberg, and charge transfer excitations or excitations with a mixture of these characters. Therefore, it is most reliable to use methods that treat all three types of excitation accurately. The last two rows of Table 1 give average mean errors in which we weighted valence, Rydberg, and charge transfer excitations each with a weight of one-third. MC-PDFT with the simple tPBE functional actually has a slightly smaller average error, when the three categories of excitation are weighted equally, than does CASPT2, but the small difference is not meaningful; what is more important is that the methods are approximately equally accurate. The MC-PDFT results are not overly sensitive to the choice of the on-top density functional (see Supporting Information, which shows that ftrevPBE is slightly more accurate). MC-PDFT is much more accurate than TD-KS-DFT with either the PBE or PBE0 exchange-correlation functional (MUE of 0.20 eV vs either 0.82 or 0.40 eV). Conclusions. In this work we benchmarked the tPBE on-top density functional for various types of vertical excitations. The tPBE functional is a first-generation approximation to the ontop density functional and is related to the PBE exchangecorrelation functional by a simple translation. Testing tPBE on vertical excitations provides a baseline on the potential usefulness of MC-PDFT for excited-state chemistry as we further improve the method.

CASPT2 calculations were also done without an IPEA shift (shift value of 0 eV), and those are reported in the Supporting Information. All calculations are done nonrelativistically. All TD-KS-DFT calculations were performed with Gaussian 09.30 Benchmarking data has been taken either from experiment or from high-level electronic structure calculations. CCSD31 geometry optimizations were performed using Gaussian 09. We used the jul-cc-pVTZ basis set for molecules with valence excitation, the aug-cc-pVTZ basis set for water, the 6-31+G** basis set for pNA and DMABN, and the aug-cc-pVDZ basis set for B-TCNE. All calculations were done using C1 symmetry.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.5b02773. Further technical information regarding computational details and further discussion of CT excitations. Excitation energies with CASSCF, CASPT2 (IPEA = 0 and 6.80 eV) and various MC-PDFT functionals for smaller active spaces. Excitation energies with various density functionals (ftPBE, trevPBE, ftrevPBE, revPBE) and CASPT2 (IPEA = 0) for moderate active spaces. (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences, under SciDAC grant no. DE-SC0008666. C.E.H. acknowledges a Newman and Lillian Bortnick fellowship.

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REFERENCES

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COMPUTATIONAL METHODS All CASSCF, CASPT2, tPBE, ftPBE, trevPBE, and ftrevPBE calculations were performed in a locally modified version of Molcas 7.9.27 All CASPT2 calculations used an imaginary shift28 of 5.44 eV to alleviate intruder states. All CASPT2 calculations in Table 1 and some of those in the Supporting Information used the standard empirical IPEA shift29 value of 6.80 eV; 589


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