Vibrational motion in the local- and normal-mode pictures

Votume 66. number 5

VIBRATIONAL t1.S MQLLER

CHEMICAL PHYSICS LETTERS

MOTION IN THE LOCAL- AND NORMAL-MODE

15

Octobci 1979

PICTURES

and 0. SONNICH BlORTENSEN

Pfz_ssits Irsrlf~ttre_ @ieme

Lirtitc?rstr_vOde?zse, Denmd

Received 29 May 1979; in final form 27 fu& 1979

Mbratronsi srzres of Hz0 are calcuhted and represcnred in both the conventionzl normal-mode (NM) and in the iodmode (LJl) picture. The St3teSare conGstently purer in the LSf than m the NM picture. The LSi picture &lows an extremely simple determinatron of ribrrt:ionalenergies and states

1. Introduction Since Lord Rayteigh‘s

fundrtmentaI work on sxnnll

vibrations in material systems, the normal-mode picture has been used almost exclusively in the discussion of vibrational motion. The reason for this is obvious. For small-amplitude vibration the potential energy can be approximated with a hamronic function and in this case the vibrational problem is exactly separsbIe in terms of the normal coordinates of the system_ Thus the complicated vibrational problem ofa large system is reduced to the simpfe probiem of a sum of simple harmonic oscillators. For a long time it has been implicitly assumed that the harmonic approximation to the molecu!ar vibrational potential was sufticient to describe the fundamental vibrational states of molecules_ Pmctiee has then been to determine normal coordinates and fundamental trequencies by diagonahsation of the total harmonic energy (i-e_ by the GF matrix method introduced by Wilson and co-workers) and to handle any small deviation from the harmonic approximation by perturbation theory_ Early work by Siebrand and co-workers [ I] and work by Henry [2] and by Swofford [3] has shown, however, that as regards highly excited vibrational states, the normalmode picture is not suitable. The reason for this is that for large-amplitude vibration the anharmonic terms in the potential energy, coupling different normal coordinates, become quite large and may easily be greater than the energy difference between

drfferent normal-mode states_ It was therefore suggested rhat the local-mode picture might be more suitable for rhe discussion ofhighly excited vibrational states. For the local-mode picture to be appropriate two conditions for the molecular vibrational energy must be fulfilled. First, the potential-energy function must be appro.ximateIy separable in a sum of local-coordinate potentials, where these potentials typically will be quite anharmonic_ Secondly the kinetic energy ccupling between different local modes must be relatively small. A quantitative test of the fust of these assumptions for the fundamentrds of the water molecule has recently been made by Elert et al, [4] _They showed that the off-diagonal potential energy matrix elements in the normal-mode picture have a larger effect than those in the local-mode picture. A discussion of the local-mode picture for water has - been given by Wallace [!5] _ He showed that an assumption of the potential energy as a sum of two Iocal-mode Morse functions was sufficient to describe all the vibrational states of the two stretching modes up to a total of 5 quanta with reasonable accuracy. He used Morse vibrational functions as a basis, but did not explicitly consider the resulting wavefunctions. Thus from Wallace’s work and also from that of Elert et al. [4] it is not directly possible to answer the fundamental question as to whether the local-mode picture, as distinct from a local-mode potential is superior to the normal-mode picture. In this note we address ourselves explicitly to this question_ We use a calculational procedure, with normal and local harmonic functions, that is more con539

gtllCJrJIip UScfUI thJn ihC GlthX trick) b&s uwd by W~lIax_ Thsreby 3 direct comparison of the wa~rfunctions in the differcnt pi-zturcs may be simpI> made. We also show_ bow rhe rcsuits of the fairly cktborxtc numerica computations can br‘ scrxliquarltit~ti~ei~ rcproduccd N it!r very littte work by using rhc anliarmanic toal-mode statLa ;1san .d~nost dizrgonrtibasiswhew only rcsonancc interxtions must be tzkcn into xcount ;IS perturbations. This appro;lcll crmbks onr’ to ;ndce 3 sinlpic aiculation not only of the \rbrrttiorrzI spectrum_ but aiso of the intemities of the difkrent \ibrrttioruI 5;lnds. Since we are interested in thr scpresentation of the vibratiorxtl wvavsfunctions for a s_Du(*cQkipotentti!, no comprrrison with euperm~ental values has been ;ittempted_

\cIlklt dlld

as regards

,\itorse

porenkd

2_ Method and results The alcuhtions are specified by the choice of the h;rmiItonian and the choice of J basis_ For the kinetic energy part of the hamilronix~ we we the convcntionrtl G matrix,

aIrhou$

rhis approach

is open

to question

Morse

highIy energ>

potential

in: vibration genad

acitcd

vibration&l

WC smpfoy ofW~11lacc

complctcl~.

quartic

stzucs.

on the one hmd [Sj which

the

rhe double

neglects the bcnd-

and on the other

putentkd

For

halld

used b> EIert et 31. [-I]

the

_

For the b&s functions we use harmonic osciliator functions but in t\\o different r\ays_ In the first cztsc w.e c1100scbnrmonic oscdhtor funcrions in tile standard normal coordinates_ In the second case we USC1o1.A arihirrmonic oscillator functions. so chosen JS to disgonalizr

the totrtl loc;lI

part of the !xm~iItonian_

The

zmharmonic oscdlator funcriorls are expressed as a linear combination oflocal harmonic oscillator functions_ For the doubk Morse potential this is equi\sIea~t to cfloosing Morse functions as basis functions but the pracnt choice is numericzdly more convenient and has more general applicability_ The b&s states in the t\\o different schemes IU+then be written 3s: (I) NormaLmode basis: i~rt) I+) Ir+_ IQ is the Irith hxmonic osciIIator function in llornraf coordinate (2)

QjLocal-mode basis: ift>lf$ll,>, LJiJ~JfWJJJoJJk

I!,> is the l,th

fUnCtiOl1 in iocid

OSdI3tOr

coordinate

and expansion densities of rhe eigsnvectors in rhr Iool-mode and rhe norm&mode pictures.for the Morse poIenri;ll. vector In~n$ indlutes ,*I quantaof the symmetric mode and “2 quanm of rbe a.q mmctric mode- In the LA1 basis subscripts s and a denote s) mmeirized and antisymmetrized skt1esrespectirely -~_____________-_-_------_. ---.-_-__-E+x~xtiua

The NM basis LB1b&S

MI basis

--

rnergy &m-l ) _~--_---_____--___I

expansion density

energy (cm-‘) _-_---

-__---_-----~

._-I_.----~-

expamion density --.-

3676 3730

0.99 I lo>, 0.99 IlO>,

3676 3730

0 85 IlO>. 0.10 120) 0.90 lOI>. 0.09 II 1)

7223 7240 7422

0.93 I20?,. 0.07 III> 098 no,, 093 II I). 0.07 I20>,

7223 7240 7422

O-44 120). 0.15 130).0.14 102).0.1 I 110). 0.09 II?) 0.59I11~.0.21 121~.0.09101) 0.62 10b. 0.18 120). O-09 112)

10.598 10600

0.98 130js

10618

032 122).0.15 1201,o-14 II?). O-IO 130,.0.10 1401

10610

035

IO899 10996

1301, 098 IX>, 0.99 ia>,

10903 10998

035 059

13798 13798

099 339

13988 13988

O-30 121). 0.14 123). 0.12 141). 0.08 0.26 112). 0.18 132). 0.12 130). 0.10

14276 I4316 1451 I

0.85 13&, OJ4 l22, 099 13113 OAS 122). 0.14 l3f)s

14333 14365 14530

0.25 SO>.021 130).0_08 II2LO.08 114). 0.07 1401 O-18 l?iL 0.18 103LO.14 113~ 0.13 123). 0.08 151) O-45 to?>. 0.15 II?). O-09 122,. 0.09 132). 0134 ISO>

099

140;s l?O&

111>.0.23 13I>.O.18 tZD.O-10 113). 0 06 141) 130,. 031 140). 0.18 l12>.01)8120~. 0.08 102) 103,. 0.15 113~ O-IO 121>.0-08 1311 103). 0.08 133) 142,. 0.05 1141

Volume

66. number 3

CHEMICAL

PHYSICS

‘ii- with

Table 2 Eigsnv~Iucs and

expansion densities of the c@nrrctors m the local-;nodr picture. !br rhc quartic po~sniiai. In rhc LX1 basis

sand n denote 5) mmrtrized

and antisymmetrized

__-.

--_-

_

Expansion densities _____--___-__

Energy (cm-t)

1637 324 1 3739 3857 -So1 5377 549?

o-99 1001) 0.97 !002) 0.99 :loo), 0.99 IlOO& 0.96 :003) O-96 IIOI~, 0.98 i101J3

6316 6975 7094 7459 7604 7722 7655 ss37 56.55 9093

0.94 0.93 0.96 0.46 0.99 052 0.95 0.90 0.94 0.45

9238 9270

0.97 l201$ 0.96 1006).

9353 10176 10218 10695 10837 10951 11167

11372 11528 11613

15

October 1979

potential, than for the qurtrtic potential. The result of the di3gonalization of the total hamiltonian IS shown in table 1 for the ,Morse potential and in tables 1 and 3 for the quartic potential_ Besides giving the energies. we display rhe expansion densities, that is the squares of the exp3nsion coefficients, in the t\\o bases- It is seen, that the resulting energies do not always agree completely for the two bases. This is due to the fact that the two bases are not equally “‘good“ and so calculations with different bases are not equally close to convergence_ For the Morse calculations, as mentioned, the bending coordinclte is ignored, and 311 states up to a total of 10 quanta are taken into account, while in the expansion of the individual Morse oscilIator functions, harmonic states up to 3 tot31 of 30 Morse

where !rri) is a 10~31 harmonic oscill3tor function_ The normal coordinates are found by diagonalizing the ilttrntonic part of the total potential, whiIe rhe expansion coefficients CT&.which express the Iocal-mode wabefunctions in terms of local harmonic functions, are found by diagonalrzmg the local part of the hamiltonian. The !Uorse potenrial of Wallace is sontewhat more anharmonic than the quartic potentiai. From what ws said earlier we should therefore expect the description of the vibration31 motion to be somewhat more tilted towards the local mode picture for the

respectircIy

LE-I-I-ERS

0.50 0.90 0.94 0.43 0.95 050 O-81

1004). I102Js. 1102), 1200),. 1700Ja I2oog. IOOSJ. i103g. 1103J, 1201J,.

12OlJ,. 1104J,. 1104J, 1702);. 1202J, 1202;;. 1210Js.

>tdtes

-

TabIe 3 Ekenwlucs and expaxion densaies of the eigenrectors in the normal-mode ptcture, for the quartic potential The NM basis hector lnt~~3J indicates 1rt quanr.t of the ~mrnetric stretch mode, jr2 quanta of the antisymmetric stretch mode and ~3 qurtnra of the symmetric bending mode

En+rs)- fcm-‘)

Expansion densities i

0.01 !003J. 0.01 1005) 0.01 1004)

6431 6979 7093

0.95 IOOIJ. 0.03 IlOlJ, 0.02 0.90 1002). 0.02 1001). 0.04 0.85 IIOOJ, 0.10 1200>. 0.03 0.90 IOlOJ, 0.09 11lOJ 0.80 1003). 0.09 1004J. 0.05 0.82 1101J.0.11 I201J.0.03 0881011J.0.091111J 0.70 1004J, 0.14 IOOSJ,0.11 0.76 1102J, O-10 i202J, 0.04 0x3 1012J.O.09 1112J.0.04

0 52 II 10)

7454 7602 7724 8112

0.64 0 66 0.81 0.66

12OOJ.O-21 IllOJ. 0.20 102OJ, 0.16 IOOSJ,0.17

8613 s722

0.69 o-77 0.61 0.64 0.79 0.65

1103). 1013). IZOlJ, II 1 lJ, 102lJ, 1006).

0.52 0.77 059 0.61 0.75 o-39 O-40 0.45 0.65

1104J, 0.18 1014). 0.09 1202J, 0.21 1112J, 0.20 1022J, 0.16 ;300>. 0.31 1310J,0.30 112OJ-0.29 103OJ, 023

1640 3243 3739 3563 4520 5367 51S8

0.47 1110) 0.02 1004J. 0.02 1006) 0 02 1005) 0.49 I1 11) 0.02

1005)

O-45 1111) 0.02 I1 13) 0 46 1112) 0 4s ii 12) O-15 13OOJ,

058 1300&. O-41 1210)~ 0.82 1300)s. 0.15 1210), 058 121O)a.O-41 1300$

9117 9254 9366 10283 10337 10417 10796 10927 11016 11252 11430 11590 11660

1002) 1003) 1000, 1007J 1001J.0.02

1102)

1003, 103OJ,O.O3 1002) 1013J.0.02 IOllJ

I30OJ. 0.10 IlOO), 0.02 1400) IZlOJ, 0.08 IOlOJ, 0.02 1030) 1120) 1004). 0.14 1006)

0.10 1203J, 0.06 1102J, 0.03 1003) 0.07 1014). O-05 1012) 0.21 13OlJ, 0.10 IIOIJ, 0.02 1202) 0.20 1211), 0.08 101lJ, 0.02 1031) 0.16 112lJ, 0.02 1112) 0.18 1104). 0.16 1005) 1006J. 0.08 1103J, 0.03 1013J. 0.07 IOlSJ, 0.07 1302). 0.09 1102J, 0.03 1212J, 0.081012), 0.03 1122). 0.03 1023). 0.02 1400). 0.19 boo,, 0.04 13lOJ, 0.15 IllOJ, O-09 1220). 0.14 102OJ. 0.03 113OJ. O-06 I1 10). 0.02

1004) 1114) 1203) 1113) 1021) 15OOJ 1030) (200) 1310)

541

3_ L%ensGorr IL is irnnx&trI> obvious tiom the faults dZspI.lyCd in t;tbIes I. 2 snd 3, rim the Iocitl-mude picrtirc is superior to the normA-mode picture. Both ttir the Bi$d_t’ zmf1rJrmonL Jforsz: po,tcntiai zlnd t!te more h;lrmotric quaatic potcntki are the w3vefunction.s purer in the lo& Lhrtn in the nomId-mode brrsis. In mauy ~;lses the ~&XL molecuiar cigenfunstiooru are in hct pmcLicAIy idcnticzl to the I_M w~vefunctions_ &en whert this is not so the IN mhins can be sunply expktincd yusIitrttiveIy-iv. Thus. it is seen that only pseudo-resonsnce intcxxtbns, that is interrtctious between stztfes with the s;tme number of IO& quanta. need to be considered_ These interactions arc of two types: the e_xact resonance inrerclcLions between states that are pennutstionliy dated (i-c states Iike !OZ> and tZO>), and the near resonance inteiacr%ons between slates Lhst in the harmonic limit would have e_uaetly the same energy (i-e_ 133 and 11I >)_ The first type of intcmction Iesds to a symmctriddtion of the Xi~efuil~t~OR, and even En tzses ffike I30). {OB) where it is numericaliy smail it should never be ignored_ since it ensures th3t the ~vavefunctions ftzve proper symmetry- In LabIes 1 and 2 symmetrization has been indicated with subscripts s and a, referring to the symrr,etricand antisymmetric components respectivefy.. The mixing In the IocaI-mode picture may be due to either kinetic energy or non-locat potential energst coup&s_ For the general quartic potenti& rend of course for the pure& focd Morse potentirtl, it is found that the kinetic energy couptings dominate_ Thus, at IeaSt for semiquantitative purposes, we czm assume that the coupIing irt the localmode picture is proportional toPLp2, wherepI is the momentum conjugate to the Ioczf coordinate 1_ When the momentum opemtor operates on harmonic osciiiator functions it has the well known step-upstep-down property, and connects only states that

differ by one quantum. Since wz are using harmonic functions as base states for the expansion of the anharmonic IouI Fiction it is easy rmmericaIIy to evzduate the effect of the momentum operrttor. Inte542

restin~l~ enough it is fi>und tbiit the basic stcp-upstep-down property is ~pI~ro~iIt~teI~ conserred e\en for quite duh.mnonxc w3vcfunct~ons. If then we, for purposes ofiiI~Istr~rio~. assume thst the m;rtrl\ elements of the momentum operator 91% iderltitlll to the harmonic matrix clemcnrs it is simple to &cubte the total energy spectrum by hand, t&irtg imo account cmiy the psqudo-resonance interactions in the foc&mode picture as preuiously discussed- This LzdcuI&ion for the I\lorsc potential is iIIustr&ed in L>g_I _Comparison with the elaborate numerical cafcuI&ions in table I shows that although there ore quantitative diffr--rences, the simpie hand tz.dcuIntion reIJroduLrs the numerical resuIts surprisingly wefi_ indeed rhe agreement Lrn be made complete if one substitutes the exact nutria elements of the momentum operator for the “harmonic” matrix elements. As regards intensities the local-mode picture may provide ;t simpIe exphnation for the fact that the observed vibrationrtf spectrum is generAlly quite narrow even for high-energy escitation. If, ;1s is usualIy done_ iL is ~sumed that electric& ttnharmonicity vanishes, then the vibmtionat operator responsible for dipole or Rztman transitions is simply the coordinate (I_ This coordinate rnzv be interpreted s 3 focal coordinate in the JAI picture & 9 normal coordinate in the NM picture.. In both pictures. tile vibrationai ground state is aImmost pure (000X Thus, in any picture, o&y matris elements to states of the types fOO&. [OaO>or f ~00) wilf be nonzero. In the IocaI-mode picture, states with a significant content of a psrticuktr state of Lhat type lie very close together- As an exampIe from tabIe I it is seen that the 0nIy states with ;t significzmt contertt of the 1201 states are separated by just IS cmWft while the corresponding separation for the threequantum excitation is2cm-l ltnd zero for the four-quarttum excitation. Thus indeed the observed vibratiomd spectrum narrows on h&her elccitation which is quite difficult to understand when one thinks in terms of the normal-mode picture. Of course using the NM picture one could numericaIIy arrive at the s3me concIusion 3s foIIows directly in the EM picture, but that would entait a discussion of very complicated interference effects between the contributions to a given band from a multitude of NM states. !t may not always be reaI&tic to restrict the v&ation of the dipole moment or the poIarWbiIity to rerms constant and Iinear in q [6] _The functiondt form of

CfIEWCAL

Volume 66. number 3

PHYSICS LETTERS

IS October 1979

4_ Conclusion

,: $30~

10606

111,

7308 --_____

___

c-b’ .‘,‘,c ‘f

L-____

10599 WE.97

7.425 __---;*. ,’

i20>

‘* ,

”

___-

.’ .*.

1239______.’

7239

--==--7221

The calculations reported here show, that at Ieast as far ASthe water molecule is concerned, the local-mode picture is certainly advantageous. It is simpier and converges much faster than the normal-mode picture; whether this will also be the case for other systems remains to be seen. It is obvious that :he large mass ratio of the atoms m water reduces the kinetrc energy couplings and so tends to make the local-mode picture purer. Also the extent to which the potential is IocsI and anharmonic is of ilttll6St importance in determining which picture is the most convenient to use. To our knowledge few potentials are known well enough from experiments to allow a detailed test, but some may become available soon from quantum chemical calculations. So far the interest in the Iccal-mode picture stems mainly from attempts to understand vibrational excirations in the electronic ground state- It seems likely, however, that the LM picture should be even more important in tryirg to understand the vibrational fine structure of electronic bands. In recent years it has often been found necessary to involve the so-called Duschinsky effect, that is normal-mode mixing, in order to expiain the intensity- distributions

in an elec-

tronic band. This effect makes the calculation ‘,D)

3701

__----------3675 ----Sp______

3732

Fig. I. Schematic representation of perturbation calculation of the vrbrationalstates of \v&ter.using the local-mode (L&i) picture- Energiesare piben in cm-t ,_nnd for the fmzd states the expansion densities in terms of LU b&s states are given. The distances between groups of states corresponding to different Icrels of excitation .ue not to sak.

these quantities then becomes important_ If it has local coordirrate structure,

thst is terms invoIvins products

of different local coordinates are smalI, then again we would arive at the same conclusion as stated above. The fact. as emphasised by Henry [2], that the phenomenological LM picture does seem to account for the intensity pattern of highIy excited vibrational states in several hydrogen-containing molecules, is an indication that the electric dipole, at leasistin these cases has Iocal coordinate structure_

of

Franck-Condon factors quite comphnted and makes simple explanations of intensity patterns impossibleThe local-mode picture could provide a new handIe on cases like that. In this connecrion it is most interesting, that in a very recent paper Wallace and MI [7] have suggested that the local-mode model may be applicable also to groups other than the relatively inactive C-H and O-H groups_ References [I 1 1%‘. Siebmnd nnd D F. Williams. J. Chem. Phys. 49 (1968) 1860. [?I B-R- Hem-y-.Accounts Chem. Res. 10 (1976) 207, and references therein. [3 1 R-L. Suofford. WE. Long and A-C. Albrecht, J. Chem.

Phys. 65 (1976) 179.

[41 M L. Elect. P.R. Stannard and WM_ GeIbart. J. Chem- Phys. 67 (1977) 5395s [S] R- Wallace, Chem. Phys. 11 (1975) 189. 161 M-S- Burberry and A-C. Albrecht. J. Chem. Phys. 70 (1978) 147. 171 R. Wallace and AA- Wu, Chem. Phys. 39 (1979) 221.

543