Lagrangian approach to molecular vibrational

THE JOURNAL OF CHEMICAL PHYSICS 126, 201104 共2007兲

Lagrangian approach to molecular vibrational Raman intensities using time-dependent hybrid density functional theory Dmitrij Rappoporta兲 and Filipp Furcheb兲 Institut für Physikalische Chemie, Universität Karlsruhe, Kaiserstraße 12, 76128 Karlsruhe, Germany

共Received 17 April 2007; accepted 3 May 2007; published online 30 May 2007兲 The authors propose a new route to vibrational Raman intensities based on analytical derivatives of a fully variational polarizability Lagrangian. The Lagrangian is constructed to recover the negative frequency-dependent polarizability of time-dependent Hartree-Fock or adiabatic 共hybrid兲 density functional theory at its stationary point. By virtue of the variational principle, first-order polarizability derivatives can be computed without using derivative molecular orbital coefficients. As a result, the intensities of all Raman-active modes within the double harmonic approximation are obtained at approximately the same cost as the frequency-dependent polarizability itself. This corresponds to a reduction of the scaling of computational expense by one power of the system size compared to a force constant calculation and to previous implementations. Since the Raman intensity calculation is independent of the harmonic force constant calculation more, computationally demanding density functionals or basis sets may be used to compute the polarizability gradient without much affecting the total time required to compute a Raman spectrum. As illustrated for fullerene C60, the present approach considerably extends the domain of molecular vibrational Raman calculations at the 共hybrid兲 density functional level. The accuracy of absolute and relative Raman intensities of benzene obtained using the PBE0 hybrid functional is assessed by comparison with experiment. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2744026兴 I. INTRODUCTION

Vibrational Raman spectroscopy is a widely used analytical tool in gas-phase and condensed-matter chemistry.1,2 Raman experiments provide a wealth of information on structure, conformations, and bonding which complements, e.g., infrared or nuclear magnetic resonance measurements. Modern spectroscopic techniques give access to wellresolved Raman spectra of larger systems such as fullerenes, nucleic acids, and proteins.3–5 Assignment of such spectra to a specific molecular structure or conformation requires theoretical simulations. In the double harmonic approximation, there are two main ingredients for the latter. 共i兲 共Harmonic兲 vibrational frequencies and normal modes; their computation is well established at various levels of sophistication.6 共ii兲 First-order nuclear derivatives of the electronic polarizability at the excitation wavelength,1,7 giving rise to Raman intensities. Here we report the first analytical implementation of first-order derivatives of molecular frequency-dependent polarizabilities in the time-dependent density functional8,9 framework. Previous work based on numerical derivatives with respect to nuclear coordinates10–12 established the usefulness of time-dependent density functional theory 共TDDFT兲 for Raman intensities and emphasized the importance of frequency dependence.11 Limitations of the finite difference approach are a linear growth of frequency-dependent polarizability calculations with system size and numerical instability. The present analytical method is based on a fully a兲

Electronic mail: [email protected] Electronic mail: [email protected]

b兲

0021-9606/2007/126共20兲/201104/5/$23.00

variational polarizability Lagrangian; analytical excited state energy calculations use similar techniques.13,14 Thus, derivative molecular orbital coefficients are avoided entirely, in contrast to previous analytical polarizability derivative implementations in the framework of time-dependent Hartree-Fock15 共TDHF兲 or static Hartree-Fock theory.16 As a result, the cost for a computation of all first-order polarizability derivatives differs only by a constant factor from the cost of a single-point polarizability calculation. The present Lagrangian approach to Raman intensities is not limited to density functional theory; it may as well be applied to correlated wave function methods, such as coupled cluster response theory 共see Ref. 17 for a general formulation of variational Fourier component perturbation theory for wave function methods兲.

II. THEORY AND IMPLEMENTATION

We define the polarizability Lagrangian Lmn关Xm,Y m,Xn,Y n,C,Zmn,Wmn兴共␻兲 = 具Xm,Y m兩共⌳ − ␻⌬兲兩Xn,Y n典 + 具Xm,Y m兩Pn,Qn典 mn + 具Pm,Qm兩Xn,Y n典 + 兺 Zia ␴Fia␴ ia␴

−

兺

pq␴,p艋q

Wmn pq␴共S pq␴

− ␦ pq兲.

共1兲

The stationary point of Lmn共␻兲 is the negative mn component of the electronic polarizability at the frequency ␻ 共m , n 苸 兵x , y , z其兲 共Ref. 18兲 within the adiabatic approximation. We 126, 201104-1

© 2007 American Institute of Physics

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J. Chem. Phys. 126, 201104 共2007兲

D. Rappoport and F. Furche

closely follow the notation of Ref. 13 and focus on the differences here. The supervectors 兩Xk , Y k典 共k is either m or n兲 contain the first-order density matrix response, expanded in the basis of products of virtual and occupied molecular orbitals and their transposes, respectively.9 The molecular orbitals 共MOs兲 ␸ p␴共r兲 are solutions of the ground state spin unrestricted Kohn-Sham 共KS兲 equations with orbital eigenvalues ␧ p␴; as usual, indices i , j , . . ., denote occupied, a , b , . . ., virtual, and p , q , . . ., general orbitals. The MOs are expanded in a basis of atom-centered contracted Gaussians ␹␮共r兲 with expansion coefficients C␮p␴. Following an idea of Helgaker and Jørgensen,19 the Lagrangian multipliers Zmn and Wmn are introduced to make the MO coefficients C satisfy the ground state KS equations.13 The superoperators ⌳ and ⌬ are ⌳=

冉 冊

A B , B A

⌬=

冉 冊 1

0

0 −1

冕

d3r␸i␴共r兲共r兲k␸a␴共r兲

共3兲

in the dipole-length gauge. The necessary condition for a stationary point of Lmn is that the functional derivative with respect to all parameters vanishes. Stationarity with respect to 具Xm , Y m兩 and 兩Xn , Y n典 leads to the time-dependent Kohn-Sham response problems,9,20,21 共⌳ − ␻⌬兲兩X ,Y 典 = − 兩P ,Q 典, k

k

k

determining the first-order vectors 兩X , Y 典. Efficient integraldirect algorithms to solve this linear problem are well established.14 Equations determining the Lagrange multipliers Zmn and mn W follow from the stationarity condition with respect to the MO coefficients C, as described in Ref. 13. The so-called Z-vector equation,

兺 共A +

jb␴⬘

=−

再兺 1 2

b

−

再

1 4

n m n ⫻共P + Q兲ab ␴ − 兺 共X + Y兲 ja␴共P + Q兲 ji␴ j

+ 共m ↔ n兲, mn Tab ␴=

再

1 4

兺i 关共X + Y兲iam ␴共X + Y兲ibn ␴ + 共X − Y兲iam ␴共X − Y兲ibn ␴兴

冎

冎

共6兲

+ 共m ↔ n兲, Here, 共m ↔ n兲 stands for the preceding expression in braces with the indices m , n interchanged. Using the linear operators H+ and H− 共Ref. 13兲 and denoting the third functional de-

兺

+ 共m ↔ n兲 + H+关Tmn兴 + 2

jb␴⬘kc␴⬙

⫻共X +

m Y兲 jb␴⬘共X

+

冎

xc

gia␴ jb␴⬘kc␴⬙

n Y兲kc␴⬙ .

共7兲

mn The relaxed one-particle density matrices Pmn pq␴ = T pq␴ quantify the first-order change of the mn polarizability component due to a change of the external potential. The remaining conditions determine the energy-weighted density matrices Wmn related to a change of the overlap matrix,

+ Zmn pq␴

1 1 + ␦ij

Wmn ij␴ =

冉再 兺 1 2

a

m n ␻关共X + Y兲ia ␴共X − Y兲 ja␴

n m + 共X − Y兲ia ␴共X + Y兲 ja␴兴 −

1 2

兺a ␧a␴关共X + Y兲iam ␴

冎

m n ⫻共X + Y兲nja␴ + 共X − Y兲ia ␴共X − Y兲 ja␴兴 + 共m ↔ n兲

+ H+ij␴关Pmn兴 + 2

冊

n

兺

kc␴⬘ld␴⬙

xc

m

gij␴kc␴⬘ld␴⬙共X + Y兲kc␴⬘

⫻共X + Y兲ld␴⬙ ,

mn Wab ␴=

共5兲

兺a 关共X + Y兲iam ␴共X + Y兲nja␴ + 共X − Y兲iam ␴共X − Y兲nja␴兴

兺j 兵共X + Y兲mja␴H+ji␴关共X + Y兲n兴 b

1 1 + ␦ab

冉再 兺 1 2

i

m n ␻关共X + Y兲ia ␴共X − Y兲ib␴

n m + 共X − Y兲ia ␴共X + Y兲ib␴兴 +

is a first-order static coupled-perturbed KS equation. The mn right-hand side Ria ␴ of Eq. 共5兲 depends only on first-order response properties including the vectors 兩Xk , Y k典 and the unrelaxed density matrices Tmn pq␴, where Tmn ij␴ = −

1 2

m + 共X + Y兲mja␴H−ji␴关共X − Y兲n兴其 + 兺 共X + Y兲ib ␴

k

mn Ria ␴,

+ m n 兵共X + Y兲ib ␴Hab␴关共X + Y兲 兴

− m n + 共X − Y兲ib ␴Hab␴关共X − Y兲 兴其

共4兲

k

k

mn B兲ia␴ jb␴⬘Z jb␴⬘

mn Ria ␴=

共2兲

,

where 共A ± B兲 are the electric and magnetic orbital rotation Hessians given in Ref. 13. The external dipole perturbation enters the Lagrangian through the vectors 兩Pk , Qk典, given by k k Pia ␴ = Qia␴ =

rivative of the exchange-correlation functional by gxc, the right-hand side becomes

1 2

兺i ␧i␴关共X + Y兲iam ␴

冎

冊

n n m ⫻共X + Y兲ib ␴ + 共X − Y兲ia␴共X − Y兲ib␴兴 + 共m ↔ n兲 ,

共8兲 mn Wia ␴=

再兺 1 2

j

兵共X + Y兲mja␴H+ji␴关共X + Y兲n兴

+ 共X − Y兲mja␴H−ji␴关共X − Y兲n兴其

冎

mn + 兺 共X + Y兲mja␴共P + Q兲nji␴ + 共m ↔ n兲 + ␧i␴Zia ␴. j

Dynamic polarizability derivatives are given by the negative derivatives of Lmn at its stationary point. Since Lmn is fully variational, computing the first-order derivative with respect to a nuclear displacement ␰ does not involve any derivatives of the parameters Xk, Y k, C, Zmn, and Wmn. Using

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J. Chem. Phys. 126, 201104 共2007兲

Vibrational Raman intensities

TABLE I. Raman scattering activities of benzene in Å4 / amu for ␻ = 20 492 cm−1 共488 nm兲, unpolarized radiation, and scattering angle of 90°. The basis sets for force constant calculations 共Basis vib.兲 and polarizability derivatives calculations 共Basis pol.兲 are specified in the table 共TZ stands for TZVPP, Sad for Sadlej’s basis set, and aTZ and aQZ denote aug-TZVPP and aug-cc-pVQZ basis sets, respectively兲. The numbering of normal modes follows Ref. 39. Experimental results are from Ref. 27. Method

PBE0

HF

Basis vib. Basis pol.

TZ TZ

TZ Sad

TZ aTZ

aQZ aQZ

␯1 共A1g兲 ␯2 共A1g兲 ␯6 共E2g兲 ␯7 共E2g兲 ␯8 共E2g兲 ␯9 共E2g兲 ␯10 共E1g兲

98.88 452.76 9.44 300.86 23.15 13.17 0.30

120.34 460.36 6.67 289.66 35.62 11.52 7.99

121.15 446.30 6.86 297.67 32.76 11.90 4.40

122.19 447.23 6.39 299.60 33.62 11.81 4.18

the variational stability and transforming to the atomic orbital 共AO兲 basis 共indicated by Greek indices兲, we obtain an expression reminiscent of the ground22 and excited state13 gradients,

⳵Lmn = 兺 h␮␰ ␯ P␮mn␯␴ − 兺 S␮␰ ␯W␮mn␯␴ + 兺 V␮xc共␯␴␰兲 P␮mn␯␴ ⳵␰ ␮␯␴ ␮␯␴ ␮␯␴ +

兺

mn

␮␯␴␬␭␴⬘

⫻共X +

共␮␯兩␬␭兲␰⌫␮␯␴␬␭␴⬘ +

Y兲␮m␯␴共X

+

n Y兲␬␭␴⬘

兺

␮␯␴

兵共rn兲␮␰ ␯共X

+ 共rm兲␮␰ ␯共X + Y兲␮n ␯␴其,

+

where h denotes the one-electron core Hamiltonian, Vxc is the ground state exchange-correlation potential, f xc is the exchange-correlation kernel in the adiabatic approximation, S is the overlap matrix, and 共␮␯ 兩 ␬␭兲 is an electron-repulsion integral in Mulliken notation. 共␰兲 indicates that derivatives are taken at fixed MO coefficients. The effective two-particle density matrix ⌫mn is given by n

⌫␮␯␴␬␭␴⬘ = P␮mn␯␴D␬␭␴⬘ + 共X + Y兲␮m␯␴共X + Y兲␬␭␴⬘ mn − 21 cx␦␴␴⬘关P␮mn␭␴D␬␯␴⬘ + P␮␬ ␴D␭␯␴⬘ n

m + 共X + Y兲␮m␭␴共X + Y兲␬␯␴⬘ + 共X + Y兲␮␬ ␴ n

n

⫻共X + Y兲␭␯␴⬘ + 共X − Y兲␮m␭␴共X − Y兲␬␯␴⬘ n

m + 共X − Y兲␮␬ ␴共X − Y兲␭␯␴⬘兴,

85.9± 8.6 368± 96 4.63± 0.93 249± 110 27.0± 5.4 9.29± 1.86 2.39± 0.49

derivatives24 along with the present analytical polarizability gradients required 3 h on a single CPU of a 2.4 GHz Opteron cluster node; the same functional and basis set as in Ref. 12 were employed. After correction for the different floating point performances 关ratio of the SpecFP CPU 2000 共Ref. 25兲 values= 7兴, this amounts to a speedup of two orders of magnitude in the total computation time. III. RESULTS FOR BENZENE

Y兲␮m␯␴ 共9兲

mn

133.51 372.33 7.51 264.97 48.27 7.11 9.36

Expt.

xc共␰兲

f ␮␯␴␬␭␴⬘

␮␯␴␬␭␴⬘

+ 2兺

TZ aTZ

共10兲

where D is the ground state KS density matrix and cx is the hybrid mixing coefficient. Equations 共5兲–共10兲 were implemented in the EGRAD module13 of the TURBOMOLE program suite.23 To illustrate the performance of our code we have computed the Raman spectrum of the buckminsterfullerene C60 at 514 nm excitation frequency. A previous implementation based on finite differences of both ground state energy gradients and of dynamic polarizabilities required ⬃14 days on six 900 MHz single-processor Athlon personal computers.12 We estimate that at least one-half of this time was spent to compute the polarizability gradient. A calculation using analytical second

Table I shows computed Raman scattering activities S26 of benzene in comparison with experiment.27 We have chosen benzene since absolute Raman scattering cross sections for all Raman-active modes are known from gas-phase measurements for this system. On the other hand, relative intensities are reported in most Raman experiments rather than absolute ones.28 We thus compare the relative Raman scattering activities S / Smax · 100 of benzene to experiment in Table II. The PBE0 hybrid functional29 was used in the computations since it yields good accuracy for frequencydependent polarizabilities30 and is less susceptible to the polarizability overestimation observed for semilocal functionals31 due to its contents of Hartree-Fock exchange. Fine numerical integration grids were employed 关size 4 共Ref. 32兲兴, along with the following basis sets: polarized triple zeta 共TZVPP兲,33 TZVPP with diffuse augmentation from Dunning’s aug-cc-pVTZ basis sets34 on nonhydrogen atoms, and Sadlej’s polarized basis sets.35 The TZVPP basis set was used for the force constant calculations since the diffuse augmentation has a little effect on the results 共but considerably increases computation times兲. Calculations using aug-ccpVQZ basis sets34,36 for both force constants and polarizability derivatives are included as near basis set limit estimates. Generally, PBE0 overestimates the absolute Raman scattering activities of benzene 共Table I兲 by 10%–15%; this has been observed for smaller molecules before.11,37 The deviations from the experimental absolute Raman scattering activities are rather uniform, though, and the relative Raman scattering activities agree with the experimental results within 1% relative to Smax 共Table II兲, the breathing mode ␯1 being a notable exception. Of particular importance is the fact that the PBE0 calculations correctly predict the ordering

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J. Chem. Phys. 126, 201104 共2007兲

D. Rappoport and F. Furche

TABLE II. Relative Raman scattering activities S / Smax · 100 of benzene for ␻ = 20 492 cm−1 共see Table I for details兲. Method

PBE0

HF

Basis vib. Basis pol.

TZ TZ

TZ Sad

TZ aTZ

aQZ aQZ

␯1 共A1g兲 ␯2 共A1g兲 ␯6 共E2g兲 ␯7 共E2g兲 ␯8 共E2g兲 ␯9 共E2g兲 ␯10 共E1g兲

21.84 100 2.09 66.45 5.11 2.91 0.07

26.14 100 1.45 62.92 7.74 2.50 1.73

27.14 100 1.54 66.70 7.34 2.67 0.99

27.32 100 1.43 66.99 7.52 2.64 0.93

of the relative intensities and thus permit a definite assignment for all Raman bands. The TDHF results, on the other hand, show considerably larger deviations 共up to 6% for relative activities excluding ␯1兲 and do not reproduce the experimental ordering of the band intensities. Diffuse augmentation of the basis sets has a distinct effect on the absolute Raman scattering activities of benzene 共Table I兲, in line with similar previous observations.37 The aug-TZVPP basis set results are in good agreement with the large aug-cc-pVQZ basis set, while the TZVPP results show sizeable deviations. The Sadlej basis shows somewhat erratic errors and incorrectly reverses the relative intensities of ␯6 and ␯1. The computed relative Raman scattering activities are less sensitive to basis set size and diffuse augmentation than the absolute intensities. The TZVPP basis set has deficiencies for the breathing mode and some weak modes but provides a reasonable overall description of the Raman spectrum.

IV. CONCLUSIONS

Frequency-dependent polarizability gradients within the TDDFT and TDHF frameworks can be computed roughly at the expense of a single-point polarizability or ground state energy calculation. This corresponds to a reduction of the scaling by one power of the system size compared to previous approaches. The key to this improvement is the elimination of derivative MO coefficients by means of the variational principle. Our stand-alone polarizability gradient code is mainly intended for Raman intensity calculations but may also be used to evaluate vibrational corrections for electronic polarizability.38 While for the simulation of Raman spectra, derivative MO coefficients have to be computed during a force constant calculation, the present method completely separates this step from the polarizability gradient calculation. Thus diffuse augmented basis sets or hybrid functionals can be used to compute Raman intensities even when this is prohibitive in a force constant calculation. Our results show that diffuse basis functions have a significant effect on Raman intensities, but little effect on harmonic vibrational frequencies, in line with previous studies.6 For benzene, TDHF Raman intensities are considerably less accurate than PBE0 Raman intensities, while the computational expense is virtu-

TZ aTZ 35.86 100 2.02 71.17 12.96 1.91 2.51

Expt. 23.34 100 1.25 67.66 7.33 2.52 0.64

ally identical. The present treatment includes Raman intensity dispersion and preresonance effects in a natural way; this will be the subject of future investigations. ACKNOWLEDGMENT

This work was supported by the Center for Functional Nanostructures 共CFN兲 of the Deutsche Forschungsgemeinschaft 共DFG兲 within Project C3.9. D. A. Long, The Raman Effect 共Wiley, Chichester, 2002兲. Handbook of Raman Spectroscopy, edited by I. R. Lewis and H. G. M. Edwards 共Marcel Dekker, New York, 2001兲. 3 M. S. Dresselhaus, G. Dresselhaus, and P. C. Eklund, J. Raman Spectrosc. 27, 351 共1996兲. 4 J. M. Benevides, S. A. Overman, and G. J. Thomas, Jr., J. Raman Spectrosc. 36, 279 共2005兲. 5 R. Tuma, J. Raman Spectrosc. 36, 307 共2005兲. 6 A. P. Scott and L. Radom, J. Phys. Chem. 100, 16502 共1996兲. 7 G. Placzek, in Handbuch der Radiologie, edited by E. Marx 共Akademische Verlagsgesellschaft, Leipzig, 1934兲, Vol. VI/2, pp. 205–374. 8 Time-Dependent Density Functional Theory, edited by M. A. L. Marques, C. A. Ullrich, F. Nogueira, A. Rubio, K. Burke, and E. K. U. Gross 共Springer, Berlin, 2006兲. 9 F. Furche, J. Chem. Phys. 114, 5982 共2001兲. 10 S. J. A. van Gisbergen, J. G. Snijders, and E. J. Baerends, Chem. Phys. Lett. 259, 599 共1996兲. 11 C. Van Caillie and R. D. Amos, Phys. Chem. Chem. Phys. 2, 2123 共2000兲. 12 J. Neugebauer, M. Reiher, C. Kind, and B. A. Hess, J. Comput. Chem. 23, 895 共2002兲. 13 F. Furche and R. Ahlrichs, J. Chem. Phys. 117, 7433 共2002兲; 121, 12772共E兲 共2004兲. 14 F. Furche and D. Rappoport, in Computational Photochemistry, edited by M. Olivucci, Computational and Theoretical Chemistry 共Elsevier, Amsterdam, 2005兲, Vol. 16, Chap. 3. 15 O. Quinet and B. Champagne, J. Chem. Phys. 115, 6293 共2001兲. 16 M. J. Frisch, Y. Yamaguchi, J. F. Gaw, H. F. Schaefer, III, and J. S. Binkley, J. Chem. Phys. 84, 531 共1986兲. 17 O. Christiansen, P. Jørgensen, and C. Hättig, Int. J. Quantum Chem. 68, 1 共1998兲. 18 The present formulation of Lmn is unsymmetric in m , n for notational 1 simplicity; in the actual implementation, the symmetrized form 2 共Lmn nm + L 兲 is used. 19 T. Helgaker and P. Jørgensen, Theor. Chim. Acta 75, 111 共1989兲. 20 M. E. Casida, in Recent Advances in Density Functional Methods, edited by D. P. Chong, Recent Advances in Computational Chemistry 共World Scientific, Singapore, 1995兲 Vol. 1, Chap. 5, pp. 155–192. 21 C. Jamorski, M. E. Casida, and D. R. Salahub, J. Chem. Phys. 104, 5134 共1996兲. 22 J. A. Pople, P. M. W. Gill, and B. G. Johnson, Chem. Phys. Lett. 199, 557 共1992兲. 23 R. Ahlrichs, M. Bär, M. Häser, H. Horn, and C. Kölmel, Chem. Phys. Lett. 162, 165 共1989兲. 1 2

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Vibrational Raman intensities

P. Deglmann, F. Furche, and R. Ahlrichs, Chem. Phys. Lett. 362, 511 共2002兲. 25 SpecFP CPU2000 data from http://www.spec.org 26 Experimental Raman scattering cross sections are usually given in the form of Raman scattering activities S = g共45共⳵␣ / ⳵Q兲2 + 7共⳵␥ / ⳵Q兲2兲 共Ref. 40兲 to facilitate comparison of measurements at different excitation wavelengths. ␣ and ␥ denote here the isotropic part and the anisotropy of the polarizability tensor ␣mn共␻兲, respectively. The polarizability derivatives are taken with respect to the mass-weighted normal modes Q of degeneracy g. 27 J. M. Fernández-Sánchez and S. Montero, J. Chem. Phys. 90, 2909 共1989兲. 28 H. W. Schrötter and H. W. Klöckner, in Raman Spectroscopy of Gases and Liquids, edited by A. Weber 共Springer, Berlin, 1979兲, Chap. 4, pp. 123–166. 29 J. P. Perdew, M. Ernzerhof, and K. Burke, J. Chem. Phys. 105, 9982

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