Frequency calculations can serve a number of different purposes:

Transcription

1 4. Frequency calculatons Frequency calculatons can serve a number of dfferent purposes: To predct the IR and Raman spectra of molecules (frequences and ntenstes. To compute force constants for a geometry optmzaton. To dentfy the nature of statonary ponts on the potental energy surface. To compute zero-pont vbraton and thermal energy correctons to total energes as well as other thermodynamc quanttes of nterest such and the enthalpy and entropy of the system. Energy calculatons and geometry optmzatons gnore the vbratons n molecular systems. In ths way, these computatons use an dealzed vew of nuclear poston. In realty, the nucle n molecules are constantly n moton. Because vbratonal motons tend to be hghly localzed wthn molecules, and the energy spacngs assocated wth ndvdual lnages tend to be reasonably smlar rrespectve of remote molecular functonalty, IR and Raman spectroscopes have a long hstory of use n structure determnaton. bratonal frequences also have other mportant uses, for example n netcs and computatonal geometry optmzaton so ther accurate predcton has been a long-standng computatonal goal. 4.. bratonal analyss Frst, consder a smple datomc molecule. The PES s one-dmensonal (datomc has only one degree of freedom - the bond length q and typcally loong somewhat le the sold curve below: q (q q 0 q As we now from physcal chemstry, the vbratonal problem for a datomc s equvalent the problem of moton of a partcle wth the reduced mass on the PES. Schrodnger equaton s:

2 d ( q ( q E ( q (4. dq For small devatons from equlbrum (mnmum the potental (q can be approxmated by a quadratc functon (parabola - the dashed curve above. Sounds famlar? It was already done above n equaton.4, except here the reference pont q0 s the mnmum: ( q ( q 0 d dq x0 ( q q 0 d dq q0 ( q q 0 (4. that means that the frst dervatve at q0 s zero, leavng only the constant and the quadratc term n 4.. To smplfy, we wll rescale the axes, so that: (q0 = 0 q0 = 0 and defne the force constant : d dq q0 Substtutng to (4. we have a Schrodnger equaton for the harmonc oscllator: d q ( q E ( q (4. dq whch has well nown solutons. In partcular, the energy levels of the lnear harmonc oscllator are quantzed as: E n n h, n = 0,,, (4.4 where h s Planc constant and s the vbratonal frequency that depends on the force constant and (reduced mass: (4.5 Ths s the frequency wth whch the molecule vbrates and s usually gven as a wavenumber ~ (4.6 c c

3 n cm - (called wavenumbers. For a polyatomc molecule wth atoms the PES, as we now, has 6 ( 5 for a lnear molecule dmensons. Typcally, the vbratonal calculatons are done n Cartesan coordnates where the surface s dmensonal. The potental energy (=(,,, where etc. are devatons from the equlbrum poston (energy mnmum s agan expanded n the Taylor seres up to the second order around the mnmum, gvng the Schrodnger equaton: ( (, E H m (4.7 n (4.7 m are actual atomc masses and H s our old frend Hessan, the by matrx of force constants: q q H H (4.8 Equaton of (4.7 s a system of coupled dfferental equatons, whch are solved, as coupled equatons usually are, by uncouplng them. That means transformng the Hessan to a new coordnate system, called normal coordnates and denoted, where t s dagonal. The transformaton s wrtten as: S, =,,, (4.9 the coupled system of equatons then becomes ndependent equatons: ( ( E, =,,, (4.0 for whch we now the solutons, because each equaton s nothng else than the Schrodnger equaton for one lnear harmonc oscllator, the same as (4..

4 Several notes about the normal coordnates and vbratonal frequences: The whole dervaton of ths secton maes sense only when the molecule s n ts energy mnmum or, alternatvely, a saddle pont. Both ensure that the frst dervatve n the Taylor expanson of energy (4. s zero. If t s not zero,.e. f the geometry s not optmzed, the whole thng goes rght out the wndow! Even though we have normal coordnates for equatons, only 6 (for a general nonlnear polyatomc are meanngful, the remanng 6 have zero vbratonal frequences, because they correspond to the translatons and rotatons of the molecule as a whole. In practce they may not be exactly zero: resdual couplngs between the external and nternal degrees of freedom are present for mperfectly optmzed structures (and n practce nothng s perfect. However, the external degrees of freedom can be proected out of the Cartesan force felds to elmnate ths problem. bratonal frequences calculated for equlbrum structures are postve real numbers, because the force constants are postve (second dervatve s postve at a mnmum of a functon. However, for saddle pont (transton states one force constant s negatve, yeldng, by equaton (4.5, an magnary frequency. Imagnary frequences (usually reported as negatve values are telltale sgns of saddle ponts! ormal mode coordnates contan the atomc masses: they are mass-weghted. Each normal mode has ts characterstc reduced mass ust le the datomc oscllator (eqn. 4.. ormal modes are complcated mxtures (combnatons of Cartesan coordnates and motons of the ndvdual atoms n the molecule. They can be delocalzed,.e. the partcular normal mode can nvolve movement of most or all atoms n the molecule. The above procedure s farly straghtforward, once the Hessan matrx s avalable. Calculaton of the Hessan,.e. the force constants,.e. the second dervatves of the energy, s the mportant and the hard part. Remember that the Hessan matrx contans x = 9 values and, even though t s symmetrc, requrng calculaton of only /(9 + values, t s stll a lot of calculaton. Above we have already dscussed analytcal vs. numercal gradents and Hessans. Frequency calculatons are farly straghtforward wth methods that have mplemented analytcal second dervatves, more tme consumng for methods wth analytcal gradents (the second dervatves can be obtaned by fnte dfferentaton of the gradents, but lmted to small molecules for methods where not even analytcal frst dervatves are mplemented. bratonal frequences are tradtonally of prmary nterest to chemsts. However, vbratonal spectra, such as IR and Raman, are not made of frequences only. Spectrum by defnton s the plot of ntensty versus frequency. That means ntenstes are also mportant, n fact, for physcal chemsts perhaps even more mportant, because more nterestng physcs s n the ntenstes than n the frequences. 4

5 Frst, as you now from P-chem, the selecton rules for the harmonc oscllator are n (4. therefore only a sngle quantum wth the energy correspondng to the vbratonal frequency, h, can be absorbed or emtted. IR absorpton ntensty for a partcular normal mode s proportonal to the change of the molecular electrc dpole moment e (don t confuse wth the reduced mass as the molecule vbrated along the normal coordnate squared: I IR e μ (4. Raman ntensty s proportonal to the change n polarzablty e wth respect to the partcular normal mode vbraton: I Raman e α (4. Ths means that for calculatons of ntenstes, dervatves of dpole moments (IR and polarzablty (Raman must also be calculated. Snce polarzablty s a hgher order property calculatng ts dervatves s more demandng than calculatng dpole dervatves: we can expect Raman calculatons to tae longer than IR calculatons (and also to be less accurate. 4.. bratonal calculatons n Gaussan Includng the Freq eyword n the route secton requests a frequency ob. The other sectons of the nput fle are the same as those we've consdered prevously. Because of the nature of the computatons nvolved, frequency calculatons are vald only at statonary ponts on the potental energy surface. Thus, frequency calculatons must be performed on optmzed structures. For ths reason, t s necessary to run a geometry optmzaton at the same level of theory (! pror to dong a frequency calculaton. It can be done by ncludng both Opt and Freq n the route secton of the ob, whch requests a geometry optmzaton followed mmedately by a frequency calculaton. However, the optmzaton performed s always a full optmzaton: constrants cannot be appled. Alternatvely, you can gve an optmzed geometry as the molecule specfcaton secton for a stand-alone frequency ob or read the optmzed geometry from the checpont fle saved from the prevous optmzaton ob. It may seem wrong to do constraned mnmzaton before frequency calculatons, snce constrans would volate the requrement of the fully optmzed structure. In some specal cases t s allowed and n fact necessary, but only wth extreme care! 5

6 Most convenently, the optmzaton and frequency obs can be lned together through --Ln-- command (see the secton.7 on mult-step obs above It s worth repeatng that a frequency ob must use the same theoretcal model and bass set as produced the optmzed geometry. Frequences computed wth a dfferent bass set or procedure have no valdty. We'll be usng the 6-G(d bass set for all of the examples and exercses n ths chapter. Ths s the smallest bass set that gves satsfactory results for frequency calculatons. For our frst example, we'll loo at the Hartree-Foc frequences for formaldehyde. Here s the route secton from the nput fle: %ch=formaldehyde.ch # RHF/6-G(d Freq Geom=AllChec Guess=Read Test Here the geometry s taen from a checpont fle that was saved from a prevous optmzaton ob on formaldehyde (at the same level of theory!. Frequences and ntenstes A frequency ob predcts the frequences, ntenstes (IR and Raman, and Raman depolarzaton ratos and scatterng actvtes for each spectral lne. ote the unts: Harmonc frequences (cm**-, IR ntenstes (KM/Mole, Raman scatterng actvtes (A**4/AMU, depolarzaton ratos for plane and unpolarzed ncdent lght, reduced masses (AMU, force constants (mdyne/a, and normal coordnates: B B A Frequences Red. masses Frc consts IR Inten Raman Actv Depolar (P Depolar (U Ths dsplay gves predcted values for the frst spectral lnes for formaldehyde. The other ones follow: A A B Frequences Red. masses Frc consts IR Inten Raman Actv Depolar (P Depolar (U The strongest IR lne s the 4 th, at 08 cm - wth the ntensty of 50 KM/Mole. 6

7 ormal Modes In addton to the frequences and ntenstes, the output also dsplays the dsplacements of the nucle correspondng to the normal mode assocated wth that spectral lne. The dsplacements are presented as YZ coordnates, n the standard orentaton: Standard orentaton: Center Atomc Atomc Coordnates (Angstroms umber umber Type Y Z The carbon and oxygen atoms are stuated on the Z-axs, and the plane of the molecule concdes wth the YZ-plane. Here s the frst normal mode for formaldehyde (t s dsplayed rght under the frequency and ntensty values for the mode : Atom A Y Z In the standard orentaton, the coordnates for all four atoms are 0. When nterpretng normal mode output, the relatve sgns and relatve values of the dsplacements for dfferent atoms are more mportant than ther exact magntudes. For ths normal mode, the two hydrogen atoms undergo the vast maorty of the vbraton, n the negatve drecton. Although the values here suggest movement below the plane of the molecule, they are to be nterpreted as moton n the opposte drecton as well. Remember that the sgns are arbtrary (.e. all pluses could be changed to mnuses and mnuses to pluses - the atoms they are oscllatng bac and forth around ther equlbrum postons. What s mportant, however, once agan, are the relatve sgns of the modes wth respect to each other (.e. f one s plus and the other mnus - they have opposte sgns. Changng the sgns wll mae plus nto mnus and vce versa, but they wll stll reman opposte! In our dagram, the moton s llustrated by showng the paths of the nucle n both drectons. Thus, the hydrogens are oscllatng above and below the plane of the molecule n ths mode. 7

8 4.. sualzng spectra and normal modes n Gabedt Mang sense of the normal mode output gets overwhelmng pretty qucly (unless you can vsualze the structure and the vbratons n your head from a long set of x, y, z coordnates and dsplacements. The best way to fgure out what the vbratons loo le s to anmate them. Agan, start wth clcng on the Dsplay Geometry/Orbtals/Densty/braton con to open the wndow, then rght clc and n the menu select Anmaton (near the bottom and braton In the new wndow (called braton, select Fle/Read/Read a Gaussan output fle. You wll see the table of the normal modes on the rght and ther graphcal representaton n the molecule wndow n the left (you may want to rotate the molecule and zoom n: You can step through the ndvdual modes to vsualze them (arrows are the dsplacements and anmate them usng the Play button. The ampltude and speed can be changed usng the Scale factor, Tme step etc. To draw the spectra, clc on Tools button at the top of the rght wndow and select Draw IR spectrum or Draw Raman spectrum. 8

9 You can manpulate the plot n varous ways: rght clc on the plot to dsplay the Menu. For example, f you want to reverse the axes, rght clc, select Render/Drectons and unchec reflect and Yreflect. 9