Chemistry Computational 5650 density functional theory
References: Erin Dahlke, "Exchange-Correlation Functionals" lecture from the "VLab Tutorial" at the U of Minnesota, 2006.
http://www.vlab.msi.umn.edu/events/download/vlab_lectures/erin/erin2.pdf
Renata Wentzcovitch, "Fundamentals of DFT" lecture from the "VLab Tutorial" at the U of Minnesota, 2006.
http://www.vlab.msi.umn.edu/events/download/vlab_lectures/renata/lecture4.pdf
MIT Course 3.320 Atomistic Computer Modeling of Materials (SMA 5107), Spring 2005. Lecture #7, "Technical Aspects of Density Functional Theory," by Nicola Marzari. Video and audio lecture is available. W. Kohn, "Nobel lecture: Electronic structure of matter wave functions and density functionals," Reviews of Modern Physics, 71(5), 1253-1266, 1999. Christopher Cramer, Essentials of Computational Chemistry, Wiley, 2002, Ch. 8. Frank Jensen, Introduction to Computational Chemistry, 2nd ed., Wiley, 2007, Ch. 6. Jorge Kohanoff, Electronic Structure Calculations for Solids and Molecules, Cambridge University Press, 2006, Ch. 4-5. Ira N. Levine, Quantum Chemistry, 5th ed., Prentice-Hall, 2000, Sect 15.20, pp 573-592. Rubn Prez, "Density Functional Theory," Universidad Autnoma de Madrid, Spain, http://metodos.fam.cie.uva.es/~doctorado/carlos/dft-ruben-v4.pdf
General Ideas DFT serves both practical and theoretical needs. Hartree-Fock extended by Configuration Interaction does well for few-electron systems such as He and H2. Electron-electron interaction complicates as N (the number of electrons) increases. A problem with the Hartree-Fock method, and correlation corrections to it, is that computational effort increases rapidly with system size. Scaling Behavior N3 N4 N5 N6 N10 Method DFT(LSDA, GGA) HF, DFT(hybrid) MP2 MP3, CISD MP7,CISDTQ
Erin Dahlke, "Exchange-Correlation Functionals," UMTC VLab Tutorial 2006
DFT.odt
1
Another problem with wave-function based methods, including Hartree-Fock theory, is that the amount of information in (3N coordinates) seems too great when compared to the amount of usable information. The usual properties are integrals over all or nearly all coordinates. For example, energy, E = (3N)*H(3N) drN electron density, n(r) = N(3N)*(3N) drN-1 (1) (2)
A theory that focuses from the start on functions of only a few variables, rather than beginning with 3N variables and then integrating over most, is appealing.
DFT (Density Functional Theory)
Walter Kohn won half of the 1998 Nobel Prize in Chemistry "for his development of the densityfunctional theory." In 1963 Kohn spent a sabbatical at the University of Paris. There, he and Pierre Hohenberg proved that the electron density determines the Hamiltonian of an electronic system, and so implies all of its properties. The next year, Kohn and his postdoctoral associate Lu Sham derived the Kohn-Sham equations at the University of California San Diego.
(Biographical material was taken from the Nobel site: http://nobelprize.org/nobel_prizes/chemistry/laureates/1998/kohn-autobio.html Les Prix Nobel. The Nobel Prizes 1998, Editor Tore Frngsmyr, [Nobel Foundation], Stockholm, 1999)
E. B. Wilson explained intuitively why electron density completely defines a system. The following is Wilson's argument as stated by Frank Jensen, Introduction to Quantum Chemistry, 2nd ed., 2007, page 232). The integral of the density defines the number of electrons. In particular, N = n(r) dr. The cusps in the density define the positions of the nuclei. The heights of the cusps define the corresponding nuclear charges.
Once the number of electrons and the types and positions of nuclei are known, the Hamiltonian can be written. From that, all wave functions and energies follow.
Rubn Prez, http://metodos.fam.cie.uva.es/~doctorado/carlo
DFT.odt
2
Hohenberg-Kohn theorems
Theorem I (Kohn called it a lemma.) The ground-state density n(r) of a bound system of interacting electrons in some external potential v(r) determines this potential uniquely. (Kohn, Rev. Mod. Phys., 1999) Kohn (Rev. Mod. Phys., 1999): "Since n(r) determines both N and v(r) ... it gives us the full H and N for the electronic system. Hence n(r) determines implicitly all properties derivable from H through the solution of the time-independent or time-dependent Schrodinger equation..." Theorem II (Kohn called it a variational principle) Given the external potential v(r) and a trial electron density ntrial(r), E = Ev[no(r)] < Ev[ntrial(r)] where no(r) is the true ground-state electron density. The minimum energy E = minntrialEv[ntrial(r)] where Ev[ntrial] = F[ntrial(r)] + ntrial(r) v(r)dr . The functional F represents the total of electron kinetic energy and electron-electron repulsion energy. The trouble with the result is that F is unknown. Density functional theory creates approximate ways of calculating F, based on its physical meaning.
Kohn-Sham Equations
The KS equations are the framework for practical DFT calculations. The KS method supposes the existence of a system of N noninteracting electrons that give the same density n(r) as the exact N-electron problem. Because the electrons do not interact, their exact wave function is a product of one-electron orbitals, the "Kohn-Sham" orbitals, i. n(r) = ii(r)2 The total energy E = Ts[n] + Ene[n] + J[n] + Exc[n] where Ts[n(r)] is the kinetic energy of a gas of non-interacting electrons at density n(r) Ene[n(r)] = vext(r) n(r) dr J[n] = n(r) n(r') |r-r'|-1 dr dr' is the "Coulomb" or "Hartree" energy Exc[n] = (T[n] Ts[n]) + (Eee[n]-J[n]), is the "exchange-correlation energy," actually a kinetic energy correction plus exchange-correlation energy minus the Coulomb energy. DFT.odt 3
The one-electron orbitals i and eigenvalues i are calculated from a one-electron Schrodingerlike equation:
[
1 - 2v eff r i = i i r 2
]
where veff(r) includes the external potential v(r), the e-e repulsion and a local (i.e., non-integral) exchange-correlation potential vxc(r) that comes from Exc[n]. v eff r = v r n r ' dr 'v xc r r -r
Kohn (Rev. Mod. Phys., 1999): "The exact effective single-particle potential veff(r) of KS theory ... can be regarded as that unique, fictitious external potential which leads, for noninteracting particles, to the same physical density n(r) as that of the interacting electrons in the physical external potential v(r)." The orbitals and eigenvalues seem similar to HF orbitals and eigenvalues and are often viewed in that way. The various approximate forms of density-functional theory come from various approximations for Exc[n] and vxc(r). (Note: Exc and vxc are not independent approximations. One can be calculated from the other.)
DFT.odt
4
Kohn-Sham and Hartree-Fock orbitals are similar.
DFT.odt
5
local density approximation (LDA)
E xc , LDA [ n]= nr xc [nr ] dr where xc[n(r)] is the exchange-correlation energy per electron. The defining characteristic of LDA is that xc depends on the value of the electron density at r, not on dn/dr. Conventionally, xc is taken from the energy of a uniform electron gas at the same density as n(r). Ceperley and Alder calculated the energy of a uniform electron gas by Monte Carlo methods in 1980. From the total energy, the exchange-correlation energy was extracted. The exchange-correlation energy is split into exchange and correlation parts.
xc = x c
The exchange energy of a uniform electron gas is known from theory. The correlation energy then comes from c = xc - x. Finally, c was fit to a parameterized equation. Sometimes, when LDA is used, the particular parameterization is also stated. A common choice is the parameterization done by Vosko, Wilk and Nusair in 1980 (referred to as VWN). Spin polarization can be included simply in density functional theory, something like the conversion of RHF to UHF. One calculates separate densities for up-spin and down-spin electrons, n(r) and n(r). Spin polarization (r),
( r )
n (r ) n (r ) n( r )
where n(r ) n (r ) n ( r )
LDA that allows spin polarization is called "LSDA." Recently, LSDA and LDA are used interchangeably. From the Gaussian 03 manual: LSDA is a synonym for SVWN. Some other software packages with DFT facilities use the equivalent of SVWN5 when "LSDA" is requested. LSDA is pretty good for solids, especially metals, but is not used much for single-molecule calculations. Its problems are: nonvariational because x and c don't exactly cancel self interaction bonds are too strong
DFT.odt
6
generalized gradient approximation, GGA
LSDA uses only the value of n(r). The GGA improves upon LDA by adding dependence on the gradient of n(r). Just as in LSDA, gradient corrections are divided into exchange and correlation corrections. GGA exchange functionals The function of Becke from 1988 ("B88" or just "B") is simple and popular. It was designed to reproduce the exchange energy of the noble gas atoms He through Rn. Perdew, Burke and Ernzerhof (Phys. Rev. Lett. 1996 and 1997) created the PBE exchange functional (and a correlation functional that has the same name) based solely on theory. PBE may be the most-used GGA exchange functional. GGA correlation functionals The two most popular GGA correlation functionals are PBE. Designed by Perdew, Burke and Ernzerhof to be used with the PBE exchange functional). The PBE functional is a joint functional for exchange and correlation. LYP. Due to Lee, Yang and Parr (Physical Review B, 37, 785, 1988). Fit to the correlation energy of Helium, with flexibility to treat more electrons.
Any exchange functional can be combined with any correlation functional. The overall exchange-correlation functional is named by appending the correlation functional's abbreviated name to the exchange functional's name. Common combinations are BLYP (B88 exchange and LYP correlation) PBE (PBE for both)
GGA results are nearly always better for molecular systems than are LDA results. However, selfinteraction continues to be a problem in GGA functionals. Exchange and Correlation energies (Hartree) for a Hydrogen atom Exact LSDA PBE BLYP 0.3125 0.2680 0.3059 0.3112 Exchange 0.0000 0.0222 0.0060 0.0000 Correlation
source: Erin Dahlke, UMn TC VLab Tutorial notes, 2006
DFT.odt
7
Hybrid Functionals
Hybrid functionals mix Hartree-Fock exchange with LSDA and GGA exchange and correlation functionals. The Hartree-Fock or "exact" exchange refers to calculating HF-type exchange integrals. The exchange energy is not the same as HF exchange energy, though, because occupied Kohn-Sham orbitals {i} are used. E HF =- xc
1 4
i j
i 1 j 2
1 r 12
i 2 j 1dr 1 dr 2
where the sums are over all electrons; i,j=1, 2, ..., N. The mixing is a linear combination using empirical coefficients. Common formulas use either one parameter or three-parameters. B3LYP: ExcB3LYP = 0.80ExLDA + 0.20ExHF + 0.72(ExB88-ExLSDA) + 0.19EcVWN + 0.81EcLYP B3LYP reference: P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch, "Ab Initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields," Journal of Physical Chemistry, 98(45), 11623-11627, 1994. B3LYP reference: Axel D. Becke, "Density-functional thermochemistry. III. The role of exact exchange," The Journal of Chemical Physics, 98(7), 5648- , 1993. "The semiempirical coefficients of [B3LYP] have been determined by a linear least-squares fit to the 56 atomization energies, 42 ionization potentials, 8 proton affinities, and the 10 first-row [H through Ne] total atomic energies ..." Atomization-test molecules were H2, LiH, BeH, CH2, CH3, etc.: first- and second-row small molecules. Ionization potentials were for atoms and small molecules. Proton affinities were for H2, C2H2, NH3 H2O, SiH4, PH3, H2S and HCl.
PBE1PBE: ExcPBE1PBE = 0.75ExPBE + 0.25ExHF + EcPBE PBE1PBE reference: Cramer, Essentials of Computational Chemistry, page 251.
Many other hybrid functionals have been published. B3LYP is probably the most commonly used for routine molecular calculations. Hybrid functionals usually give better results than either LSDA or GGA functionals, but not always.
DFT.odt
8
DFT functional comparison for water molecule and dimer. Data are from Erin Dahlke's UMTC Vlab Tutorial. Basis sets unknown. water monomer R(O-H) () (H-O-H) Hartree Fock 0.9399 106.3 LSDA 0.9701 105.0 GGA PBE 0.9690 104.2 GGA BLYP 0.9706 103.5 hybrid PBE1PBE 0.9575 104.9 hybrid B3LYP 0.9604 105.1 Hybrid functionals PBE1PBE and B3LYP give good bond lengths and angles. LSDA and GGA give bonds that are too long, HF bonds are too short.
Hybrid functionals give geometries intermediate between the results of HartreeFock theory and pure density functional theory.
Dimer binding energy (kJ/mol) water dimer Ebind(kJ/mol) Hartree Fock 15 LSDA 33 GGA PBE 21 GGA BLYP 17 hybrid PBE1PBE 21 hybrid B3LYP 19 "accurate" 21 Dimer binding energy, involving a hydrogen bond, is treated better by GGA and hybrid functionals than by HF and LSDA. DFT.odt 9
The B3LYP functional is good for most chemical energies and bond lengths. HF and LSDA should not be used for reaction energies, if bonds are broken. MP2 and GGA calculations may be satisfactory. Conformational energies are good with B3LYP. Mean absolute errors Method Bond Lengths1 Atomization2 Conformers3 Molecular Orbital Angstroms kJ/mol kJ/mol HF 0.022 310 3 MP2 0.014 31 2 QCISD 0.013 na na DFT LSDA SVWN 0.017 150 4 DFT GGA BLYP 0.014 21 na PBE 0.012 36 na DFT hybrid B3LYP 0.004 9.2 2 1. Bond lengths (in Angstroms) were calculated for 32 molecules containing only first-row atoms (a subset of the G2 test set). Basis set was 6-311G(d,p). Reference: Cramer, Essentials of Computational Chemistry, Table 8.5. QCISD is like CISD plus some higher excitations 2. Atomization energies (kJ/mol) were calculated for the G2 set: 55 molecules including first- and second-row atoms. Basis set was 6-311+G(3df,2p). Reference: Cramer, Essentials of Computational Chemistry, Table 8.1. For SVWN the basis set was only 6-31G(d). 3. Energy differences (kJ/mol) for 8 pairs of conformers of organic compounds. Basis set 6-31G*. Reference: Levine, Quantum Chemistry, 5th ed., Sect 17.2.
DFT.odt
10
Time-dependent density functional theory
References: Furche, Fillip; Ahlrichs, Reinhart.Adiabatic time-dependent density functional methods for excited state properties. The Journal of Chemical Physics 2002, 117(16), 7433-7447. Marques, M. A. L.; Gross, E. K. U. Time-dependent density functional theory. Annual Reviews of Physical Chemistry 2004, 55, 427-455. Burke, Kieron; Werschnik, Jan; Gross, E. K. U. Time-dependent density functional theory: past, present and future. The Journal of Chemical Physics 2005, 123, 062206-1 062206-9. Furche, Fillip; Burke, Kieron. Time-dependent density functional theory in quantum chemistry. Annual Reports in Computational Chemistry 2005, 1, Chapter 2, 19-30. General Idea In ordinary ground-state DFT, one-particle Kohn-Sham equations produce the electron density n(r) from the fictitious external potential vs(r). The density will be the same as the correct manyelectron density if vs(r) can be chosen correctly. Recall that vs contains two known parts, the external Coulomb potential due to nuclei and the Hartree potential, and the unknown exchangecorrelation potential. Time-dependent density functional theory, TDDFT, defines "a set of timedependent Kohn-Sham (TDKS) equations that reproduce n(r,t) from a TDKS potential. This consists of the external potential, the Hartree potential, and the unknown time-dependent exchange correlation (XC) potential vxc[n](r,t)." (Furche and Burke, 2005, page 20). Approximations to vxc lead to approximate time-dependent density n(r,t). The time-dependent density can be calculated directly or, as is common in chemistry, studied with perturbation theory.
Review of stationary DFT and extension to TDDFT (Marques and Gross, 2004) In 1964, Hohenberg and Kohn proved that "to fully describe a stationary electronic system it is sufficient to know its ground-state density." Density is a good target of theory because it is observable, intuitive and depends on only three spatial coordinates. Hohenberg and Kohn established a variational principle: E[n] is minimal at the exact n(r). Kohn and Sham proposed the auxiliary non-interacting system (the "KS" system) that has the same electron density as the real, interacting system. The KS system has an effective external potential vS. vS(r) = vext(r) + vHartree(r) + vxc(r) (1)
DFT.odt
11
Many-body effects are in vxc(r). vHartree is the Coulomb energy. Kohn and Sham introduced orbitals that satisfy one-electron equations and, squared, sum to the total electron density. Time-dependent density functional theory, in its modern form, was started in 1984 by Runge and Gross. They showed that there exists a time-dependent Kohn-Sham potential vS(r,t) that, in a time-dependent one-electron equation, reproduces the exact time-dependent electron density n(r,t). As in stationary DFT, vS(r,t) is a functional of the density. It also depends on the initial state, at t=0. It also depends on the current, if there is a current, but that is not a problem for calculations on isolated molecules. Runge and Gross worked initially on scattering. TDDFT is now commonly used to calculate electronic spectra.
TDDFT Theory
Marques and Gross, 2004, stated: "The central theorem of TDDFT (the Runge-Gross theorem) proves that there is a one-to-one correspondence between the external (time-dependent) potential, Vext(r,t) and the electronic density n(r,t) for many-body systems evolving from a fixed initial state." nr ,t v ext r ,t H r ,t so knowing n(r,t) or Vext is equivalent to knowing , in principle. As in stationary DFT, the electron density is represented by a sum over one-electron Kohn-Sham orbitals. nr ,t = j r , t 2
j
(2)
(3)
Each orbital is the solution of a time-dependent Kohn-Sham equation (TDKS):
i d j r , t dt 1 = - 2v s [n]r , t j r , t , 2
in atomic units
(4)
With the proper (unknown) choice of the reference system's potential vs(r,t), the {j} give the exact many-electron density n(r,t). v s r , t = v ext r ,t n r ,t 3 d r v xc r ,t r -r (5)
The exchange-correlation potential depends on position and on time. In the limit t0, the timedependent problem approaches the zero-time stationary initial state so vxc(r,0) should be the same as an ordinary stationary DFT exchange-correlation potential. Little is known about time dependence of vxc. Most commonly, the time dependence is neglected. That is the "adiabatic" approximation. Adiabatic approximation: = vxc[n](r,t) vxcgrnd state[n0](r) where n0 equals n(r,t). (6)
DFT.odt
12
The electron density n(r) is allowed to vary with time. At any instant, vxc is taken to be the same as the stationary ground-state vxc at that density. Burke, Werschnik and Gross, 2005, wrote that "Any ground-state approximation (LDA, GGA, hybrid) automatically provides an adiabatic approximation (e.g., ALDA) in TDDFT." Many TDDFT calculations use ALDA (adiabatic local density approximation) exchange-correlation. Adiabatic B3LYP is popular in chemistry. The adiabatic approximation gives one a recipe for vxc(r,t), so n(r,t) can in principle be calculated. How n is calculated and what is done with n depends on the target of the calculation. Burke, Werschnik and Gross divided TDDFT applications into three categories: Case i. Optical excitation and electronic transitions can be treated with linear-response theory and perturbation theory. The density responds linearly to a spatially uniform field. The field couples to the electrons in the dipole approximation. Both frequency and intensity can be predicted. For these calculations, TDDFT is becoming a standard tool in chemistry. Case ii. systems in strong laser fields. Electric field strength is similar to the electronnucleus Coulomb field. TDDFT can calculate changes in (r,t). Perturbation theory fails. Case iii. Ground-state properties can be calculated using TDDFT. This is still uncommon.
Other types of calculations: geometries and other properties of excited states solid-state properties charge transfer, both intermolecular and intramolecular solvation circular dichroism
Let us consider only one application: electronic excitation and optical spectra.
Excitation energies from TDDFT
One way to calculate transition energies is to calculate the energy of the ground state, then the energy of the excited state, and take the difference. That is a "SCF" calculation. If the excited state and the ground state are at the same geometry (as suggested by the Franck-Condon principle) then the excitation energy is "vertical." If both states are relaxed to their equilibrium geometries, the excitation energy is "adiabatic." Experiments and calculations are usually for vertical excitation.
DFT.odt
13
Calculating excited states is possible in DFT but not as easy as calculating ground states. Exchange-correlation potentials are better known and optimized for ground states. The TDDFT approach is to calculate excitation energies from changes in the ground-state electron density. TDDFT uses the ground-state electron density and external potential. A change in the external potential causes a change in the electron density. The density change is connected to the potential change through the "linear density response function," , by definition of . ( is also called the "susceptibility.") n(r,t) = (r,t,r,t) vext(r,t) d3r (7)
Because the goal is excitation frequencies (as in a spectrum) it is customary to change from time to frequency, by Fourier transform. n(r,) is the transform of n(r,t)-n(r,0), and (r,r,) is the transform of (r,t,r,t) from (t-t) to . vext can be written as a function of frequency (e.g., the exciting radiation's frequency). n(r,) = (r,r,) vext(r,) d3r Equivalently, (r,r,) = n(r,)/vext(r,) (8) (9)
(Note: Similar equations appear in the references, but some include spin variables or . For simplicity, spin variables are suppressed here, as would be appropriate for a closed-shell singlet calculation.) Excitation frequencies are resonant frequencies. They are values for which n(r,) is large. In terms of the integral expression above, resonant frequencies are poles of (r,r,). Unfortunately, is unknown. The TDDFT approach is to write as a Kohn-Sham s plus corrections. The density change can be written (in principle exactly) in terms of the Kohn-Sham potential. By definition of s, the change in density is the same whether caused by vext or vs. (Compare to equation 8.) n(r,) = s(r,r,) vs(r,) d3r (10)
The integrands of equations 8 and 10 must be equal. That equality can be used to express the true in terms of the Kohn-Sham s. The result is given in equation 11. r , r ' , = s r , r ' , s r , r ' ' ' ,
[
1 f r ' ' ' ,r ' ' , n r ' ' , r ' , d 3 r ' ' ' d 3 r ' ' r ' '-r ' ' ' xc
]
(11)
Equation 11 shows that the true linear response function equals s plus a correction. The correction is nontrivial, because it involves the frequency-dependent exchange-correlation functional fxc and itself. The leading term in , the Kohn-Sham response functional s, can be DFT.odt 14
calculated from Kohn-Sham orbitals and perturbation theory (Burke, Werschnik and Gross, 2005). The poles of s are simply Kohn-Sham orbital energy differences, ia. The true excitation frequencies n will be similar to ia (as is similar to s) but corrected for exchange and correlation. The true and its n values could be obtained from the integral equation 11, but for practical calculations the integral equation was recast as a matrix equation (Casida, 1996, see Furche and Burche, 2005). Casida's equations yield the excitation frequencies n as eigenvalues. The accompanying eigenvectors show which Kohn-Sham orbitals are involved in the transitions. The eigenvectors also can be used to calculate absorption intensities.
Approximations
The theory is so far exact, within linear-response perturbation theory, which requires that the electron density change not too much from the initial ground-state density. In practice, there are three sorts of approximations that limit accuracy of results. 1. The ground-state exchange-correlation functional is not exactly known. LDA, GGA, meta and hybrid functionals may be good, but they are not exact, so the stationary t=0 electron density is not exact. 2. The frequency-dependent exchange-correlation functional fxc(r,r,) is not known. Ordinarily, the "adiabatic approximation" is used. Adiabatic vxc[n](r,t) = vxc,ground[n=n(r,t)](r) where vxc,ground is an ordinary ground-state exchange-correlation potential. The adiabatic fxc does not depend on . Quoting Burke, Weschnik and Gross, 2005, "Any ground-state approximation (LDA, GGA, hybrid) automatically provides an adiabatic approximation (e.g., ALDA) in TDDFT." "ALDA," for example, stands for "adiabatic local density approximation." Adiabatic B3LYP seems common in chemical publications. Fortunately, it appears that choice of ground-state functional is more important than choice of fxc (Marques and Gross 2004), at least for excitation energies of molecules. 3. Single excitations are included in Casida's equations, but multiple excitations are not. Multiple excitations are not treated by the usual TDDFT methods.
DFT.odt
15
Results
Atomic excitation Petersilka, 1996, compared n values (poles of ) to ia values (poles of s) and to experimental transition energies, for 1S1P excitations of several atoms. Their data are below, as given by Marques and Gross (2004, Table 1). Energies are in Hartree. The local density approximation was used for the stationary state and ALDA was used for fxc. Overall, correcting ia to n improved excitation energies. Atom Be Mg Ca Zn ia n Expt %error ia %error n 34 22 19 17 3 10 22 12
0.129 0.200 0.194 0.125 0.176 0.160 0.088 0.132 0.108 0.176 0.239 0.213
Marques and Gross suggested that much of the error in n is due to the ground-state local density approximation.
DFT.odt
16
Vertical excitation Molecular electronic excitation energies are usually calculated at the equilibrium geometry of the ground-state molecule. Such excitations are called "vertical."
Cyanine dyes.
A typical study of vertical excitation, leading to interpretation of uv-visible absorption spectra, is that of Benoit Champagne, et al., "TDDFT investigation of the optical properties of cyanine dyes, Chemical Physics Letters, 2006, 425,105-109. Consider one of the dyes (#4) they studied. The experimental spectrum was taken of a dilute solution in methanol. Champagne optimized the geometry using DFT with the B3LYP hybrid functional and the 6-311G* basis set. Vertical excitation energies were calculated with TDDFT. For comparison, both B3LYP (20% HF exact exchange) and PBE0 (25% HF exact exchange) were used with TDDFT. TDDFT xc functional and basis set B3LYP/6-311G* B3LYP/6-311+G* PBE0/6-311G* B3LYP/6-311G* methanol solvent correction Experiment E (eV) 2.75 2.74 2.79 2.62 2.29 (nm) 451 453 444 473 542
Adding diffuse functions had little effect, as expected when the orbitals involved are not highly excited. Changing from B3LYP to PBE0 also had little effect. The solvent correction (which was the "integral equation formalism," IEF, version of the "polarizable continuum model," PCM) moved E in the right direction. TDDFT overestimated excitation energy by 0.3-0.4 eV. Burke, Werschnik and Gross, 2005, suggested that TDDFT vertical excitations usually are within 0.4 eV of experiment (=60 nm at =400nm corresponding to E=3.1eV) so Champagne's theory/experiment difference is typical of TDDFT. Champagne, et al., went on to calculate an absorption spectrum. To calculate a spectrum, they Changed functional and basis set to BHandHYLP and SV(P) Shifted E down by 0.5 eV, to match experimental max Calculated vibration frequencies and Franck-Condon factors. Broadened each vibrational transition as a Gaussian with FWHM 0.05eV. Summed all the broadened vibrational transitions.
DFT.odt
17
retinal
Carlos Silva Lopez, et al., calculated vertical excitation energies of retinal and related molecules using several methods and compared results. ("Computation of vertical excitation energies of retinal and analogs: scope and limitations," Journal of Computational Chemistry, 2005, 27(1), 116-123) Ground-state geometries were optimized with B3LYP 6-31+G(d). Vertical excitation energies were calculated with many methods, and the 6-31++G(d,p) basis set. The table shows the lowest excitation energy (and wavelength) for retinal, as calculated with CI singles and five TDDFT methods. Also tabulated are the mean signed errors (MSE) in and in E over retinal and 16 analogs. TDDFT GGA BLYP 3.25 381 -0.19 +31 TDDFT mGGA SVXC 3.40 364 -0.07 +11 TDDFT B3LYP 20%HFx 2.94 421 -0.06 +8 TDDFT BH&H 50%HFx 3.40 365 +0.32 -46
CIS E(eV) (nm) 4.07 304
TDDFT LDA 3.23 383 -0.20 +31
Expt 3.25 381 NA NA
MSE (eV) +0.97 MSE (nm) -110
On average, over the group of 17 molecules, CIS gives E too large by approximately 1eV. Over the whole group of molecules, pure density functionals, LDA and GGA and the metaGGA, underestimate E. Increasing percentage HF exact exchange raises excitation energy.
DFT.odt
18
Adiabatic excitation Adiabatic excitation is from the equilibrium geometry of the ground electronic state to the equilibrium geometry of the excited state. Calculating adiabatic excitation requires selection of an electronic excited state and geometry-optimization of that state, which TDDFT can do. Furche and Ahlrich (2002) gave a computational method and tabulated comparative results. They used an augmented (added diffuse functions) triple-zeta-valence doubly polarized basis set, augTZVPP. They found TDDFT excited-state geometries ...