Computer Laboratory DFT and Frequencies
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1 Computer Laboratory DFT and Frequencies 1. DFT KEYWORDS FOR DFT METHODS Names for the various pure DFT models are given by combining the names for the exchange and correlation functionals. In some cases, standard synonyms used in the field are also available as keywords. Exchange Functionals. The following exchange functionals are available in Gaussian 03: Slater: ρ 4/3 with theoretical coefficient of 2/3, also referred to as Local Spin Density exchange [75,76,77]. Keyword: Used Alone: HFS, Comb. Form: S Xαρ 4/3 with the empirical coefficient of 0.7, usually used when this exchange functional is used without a correlation functional [75,76,77]. Keyword: Used Alone: XAlpha, Comb. Form: XA. Becke 88: Becke's 1988 functional, which includes the Slater exchange along with corrections involving the gradient of the density [462]. Keyword: Used Alone: HFB, Comb.Form: B. Perdew-Wang 91: The exchange component of Perdew and Wang's 1991 functional [463,464,465,466,467]. Keyword: Used Alone: N/A, Comb. Form: PW91. Barone's Modified PW91: The Perdew-Wang 1991 exchange functional as modified by Adamo and Barone [468]. Keyword: Used Alone: N/A, Comb. Form: MPW. Gill 96: The 1996 exchange functional of Gill [469,470]. Keyword: Used Alone: N/A, Comb. Form: G96. PBE: The 1996 functional of Perdew, Burke and Ernzerhof [471,472]. Keyword: Used Alone: N/A, Comb. Form: PBE. MPBE: Adamo and Barone's modification of PBE [473]. Alone: N/A, Comb. Form: MPBE. OPTX: Handy's OPTX modification of Becke's exchange functional [474]. Keyword: Comb. Form: O. The combination forms are used when one of these exchange functionals is used in combination with a correlation functional (see below).
2 Correlation Functionals. The following correlation functionals are available, listed by their corresponding keyword component: VWN: Vosko, Wilk, and Nusair 1980 correlation functional(iii) fitting the RPA solution to the uniform electron gas, often referred to as Local Spin Density (LSD) correlation [475] (functional III in the paper). VWN V(VWN5): Functional V from the 1980 paper which fits the Ceperly-Alder solution to the uniform electron gas (this is the functional recommended in the paper) [475]. LYP: The correlation functional of Lee, Yang, and Parr which includes both local and non-local terms [476,477]. PL (Perdew Local): The local (non-gradient corrected) functional of Perdew (1981) [478]. P86 (Perdew 86): The gradient corrections of Perdew, along with his 1981 local correlation functional [479]. PW91 (Perdew/Wang 91): Perdew and Wang's 1991 gradient-corrected correlation functional [463,464,465,466,467]. B95 (Becke 95): Becke's τ-dependent gradient-corrected correlation functional (defined as part of his one parameter hybrid functional [480]. PBE: The 1996 gradient-corrected correlation functional of Perdew, Burke and Ernzerhof [471,472]. MPBE: Adamo and Barone's modification of PBE [473]. All of the keywords for these correlation functionals must be combined with the keyword for the desired exchange functional. For example, BLYP requests the Becke exchange functional and the LYP correlation functional. SVWN requests the Slater exchange and the VWN correlation functional, and is known in the literature by its synonym LSDA (Local Spin Density Approximation). LSDA is a synonym for SVWN. Some other software packages with DFT facilities use the equivalent of SVWN5 when "LSDA" is requested. Check the documentation carefully for all packages when making comparisons. Correlation Functional Variations. The following correlation functionals combine local and non-local terms from different correlation functionals: VP86: VWN5 local and P86 non-local correlation functional. V5LYP: VWN5 local and LYP non-local correlation functional.
3 Standalone Functionals. The following functionals are self-contained and are not combined with any other functional keyword components: VSXC: van Voorhis and Scuseria's τ-dependant gradient-corrected correlation functional [481]. HCTH/*: Handy's family functional including gradient-corrected correlation [482,483,484]. HCTH refers to HCTH/407, HCTH93 to HCTH/93, HCTH147 to HCTH/147, and HCTH407 to HCTH/407. Note that the related HCTH/120 functional is not implemented. Hybrid Functionals. Three hybrid functionals, which include a mixture of Hartree-Fock exchange with DFT exchange-correlation, are available via keywords: Becke Three Parameter Hybrid Functionals. These functionals have the form devised by Becke in 1993 [79]: A*E Slater X +(1-A)*E HF X +B*ΔE Becke X +E VWN non-local C +C*ΔE C where A, B, and C are the constants determined by Becke via fitting to the G1 molecule set. There are several variations of this hybrid functional. B3LYP uses the non-local correlation provided by the LYP expression, and VWN functional III for local correlation (not functional V). Note that since LYP includes both local and nonlocal terms, the correlation functional used is actually: C*E LYP VWN C +(1-C)*E C In other words, VWN is used to provide the excess local correlation required, since LYP contains a local term essentially equivalent to VWN. B3P86 specifies the same functional with the non-local correlation provided by Perdew 86, and B3PW91 specifies this functional with the non-local correlation provided by Perdew/Wang 91. Becke One Parameter Hybrid Functionals. The B1B95 keyword is used to specify Becke's one-parameter hybrid functional as defined in the original paper [480]. The program also provides other, similar one parameter hybrid functionals, as implemented by Adamo and Barone [480,485]. In one variation, B1LYP, the LYP correlation functional is used (as described for B3LYP above). Another version, MPW1PW91, uses modified Perdew-Wang exchange and Perdew-Wang 91 correlation [468 ]. Becke's 1998 revisions to B97 [486,487]. The keyword is B98, and it implements equation 2c in reference [487]. Handy, Tozer and coworkers modification to B97: B971 [482]. Wilson, Bradley and Tozer's modification to B97: B972 [488]. The 1997 hybrid functional of Perdew, Burke and Ernzerhof [472]. The keyword is PBE1PBE. This functional uses 25% exchange and 75% correlation weighting. Half-and-half Functionals, which implement the following functionals: BHandH: 0.5*E HF X + 0.5*E LSDA LYP X + E C BHandHLYP: 0.5*E HF X + 0.5*E LSDA X + 0.5*ΔE Becke88 LYP X + E C
4 ACCURACY CONSIDERATIONS A DFT calculation adds an additional step to each major phase of a Hartree-Fock calculation. This step is a numerical integration of the functional (or various derivatives of the functional). Thus in addition to the sources of numerical error in Hartree-Fock calculations (integral accuracy, SCF convergence, CPHF convergence), the accuracy of DFT calculations also depends on number of points used in the numerical integration. The "fine" integration grid (corresponding to Integral=FineGrid) is the default in Gaussian 03. This grid greatly enhances calculation accuracy at minimal additional cost. We do not recommend using any smaller grid in production DFT calculations. Note also that it is important to use the same grid for all calculations where you intend to compare energies (e.g., computing energy differences, heats of formation, and so on). Larger grids are available when needed (e.g. tight optimization of certain kinds of systems). An alternate grid may be selected by including Integral=(Grid=N) in the route section (see the discussion of the Integral keyword for details). 2. Frequencies This calculation type keyword computes force constants and the resulting vibrational frequencies. Intensities are also computed. By default, the force constants are determined analytically if possible (for RHF, UHF, MP2, CIS, all DFT methods, and CASSCF), by single numerical differentiation for methods for which only first derivatives are available (MP3, MP4(SDQ), CID, CISD, CCD, QCISD, and all semi-empirical methods), and by double numerical differentiation for those methods for which only energies are available. Vibrational frequencies are computed by determining the second derivatives of the energy with respect to the Cartesian nuclear coordinates and then transforming to massweighted coordinates. This transformation is only valid at a stationary point! Thus, it is meaningless to compute frequencies at any geometry other than a stationary point for the method used for frequency determination. The recommended practice is to compute frequencies following a previous geometry optimization using the same method. This may be accomplished automatically by specifying both Opt and Freq within the route section for a job.
5 Specifying #P in the route section produces some additional output for frequency calculations. Of most importance are the polarizability and hyperpolarizability tensors (they still may be found in the archive entry in normal print-level jobs). They are presented in lower triangular and lower tetrahedral order, respectively (i.e., α XX,α XY, α YY, α XZ, α YZ,α ZZ and β XXX, β XXY, β XYY, β YYY, β XXZ, β XYZ, β YYZ, β XZZ, β YZZ, β ZZZ ), in the standard orientation: Dipole = D D D-01 Polarizability= D D D D D D+00 HyperPolar = D D D D D D D D D D+01 #P also produces a bar-graph of the simulated spectra for small cases. 3. Vibrations in GaussView Displaying Vibrational Modes and Spectra The Results=>Vibrations menu item is used to access various spectra results (except NMR). It allows calculated vibrational data to be displayed as dynamic screen motions, based on information from a frequency calculation. This menu item displays the Display Vibrations window, as illustrated in Figure 64. The Display Vibration dialog box shows which vibration is currently selected for dynamic display. You can start this display by selecting the Start button, and halt it by clicking on the Stop button. The molecule will begin a cyclical displacement showing the motions corresponding to the vibration selected. You can select other modes by selecting the desired mode on the scrolling list in the Display Vibrations dialog. You can select a new mode without halting the previous one. You can also rotate and move the vibrating molecule, using the normal mouse buttons. Structure and frequencies of F 3 - Method R Sym.stretch Bend Asymm.stretch LSDA MP B3LYP Exp 440 +/ / /-20
6 MP2: %chk=c:\documents and Settings\dmitry\My Documents\f3mp2.chk %mem=6mw %nproc=1 # opt freq rmp2/lanl2dz geom=connectivity F3- Mp2-1 1 F F 1 B1 F 1 B2 2 A1 B B A B3LYP: %chk=c:\documents and Settings\dmitry\My Documents\f3b3lyp.chk %mem=6mw %nproc=1 # opt freq rb3lyp geom=connectivity d95v+(d) F3- B3LYP -1 1 F F 1 B1 F 1 B2 2 A1 B B A
7 Method Scale Frequenc Factor ZPE/Therm HF/3-21G l HF/6-31G(d) MP2(Full)/ MP2(FC)/ SVWN/6-31G(d) BLYP/6-31G(d) B3LYP/6-31G(d) Computed values of the intensities should not be taken too literally. However, tit j relative values of the intensities for each frequency may be reliably compared. Scaling Frequencies and Zero-Point Energies Frequencies computed with methods other than Hartree-Fock are also scaled to similarly eliminate known systematic errors in calculated frequencies. The followng table lists the recommended scale factors for frequencies and for zero-point energies and for use in computing thermal energy corrections (the latter two items are discussed later in this chapter), for several important calculation types. As the table indicates, the optimal scaling factors for the frequencies themselves and for the zero-point energies and for use in computing thermal energy corrections are slightly different. However, it is also common practice to use the same factor for both of them ( in the case of Hartree-Fock). For example, the G2 high accuracy energy method scales computed HF/6-31G(d) zero-point energy corrections by (see Chapter 7). You should be aware that the optimal scaling factors vary by basis set. For example, Bauschlicher and Partridge computed the B3LYP/6-311+G(3df,2p) ZPE/thermal energy correction scaling factor to be * Additional scaling factors have also been computed by Wong and by Scott and Radom. Consult the references for detailed discussions of these issues. Most of the scale factors in this table are from the recent paper of Wong. The HF/6-31G(d) and MP2(Full) scale factors are the traditional ones computed by Pople and coworkers and cited by Wong. Note that the MP2 scale factor used in this book is the one for MP2(Full) even though our jobs are run using the (default) frozen core approximation. Scott and Radom computed the MP2(FC) and HF/3-21G entries in the table, but this work came to our attention only just as this book was going to press. Their value is for the 6-31G(d) basis set. Note that published scale factors often vary slightly from one another due primarily to differences in the molecule sets used to compute them.
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