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Improved semiclassical calculation of energy expectation values for diatomic molecules
2 Apnl J98f
CHEMICAL PHYSICS LE-JTERS
Votume 87. number 4
IMPROVED SEMICLASSICALCALCULATION OF ENERGY EXPECTATION VALUES FOR DIATOMIC MOLECULES PA. VICHARELL~
and C.B. COLLINS
Recewed 28 Scptcmber 1981; in tinal form 1.1Januxy 1982
Scmichsslwl e\prewonr for fhc c\pecr.Mon VJIW of the kinctrc and potcntul cncr&ys of ;1 dlJtOmlc malcculc 311: denvcd using 3 modrficd quanruation condrrlon Apphatron of this trcxmcm to rhc 1Xi state oi 111results m Improwti numcr~uI3grc~mcnt
wtrf,
the accurate
qo3nIal e\pccratlon
values
written in the form
I. Introduction The ctasstcal expectation
value of the kinetic energy of a dialonlic molecule IS given m atomic units by
rhe ehpressmn, (T) 3 $kdr/$dl
0)
= ~~~~I~~~P)~,
(3
tr>=4(u+~)dEld(u+f)=IdEldIn(u+~).
(8)
This expression compared very well with accurate
results [2] for H2, although small drfferences could be observed. Following a different treatment, Tippmg [3] obtained more precise expressions for (TI and (19 m the low ulimtt, by using the D~ham oscrktor 3pproxlmation His results were excellent uptolJ=3.
where P= {2p[E-
U(R)])“_‘,
(3)
U(R) = V(R) + X&R’,
(4)
and X=J(J+
1).
6)
Here T, V, and E are the kinetic, potenrial, and total energies, respecttvely, and ~1is the nuckar reduced IIIWS. The expectation value of the potential energy is obtained from
(13=E-
Ul.
(6)
the especratron vah~eof the
(7) kinetic
simple anslytical result obtained from the latter
present deriv3tion differs from earher the modified quantiz3tion condition takesinto account the f3ct thst the potential is defined on th‘e (0, m) semi-infirute interval, and not on mAod.
work
The
in that
the (--,
-) infirute
interval
requwec!
for cq. (7). As
expected, the improved equations dimimsh the prcvlousiy observed g3p between semrclassica1and quantal results, and their suttpkity makes them very at-
Stwalley [l] has shown that by using the firstorder semiclassical quantization condition (u f I) = (I/~~)~~~,
In prmciple, more accurate results may be obtsined for all v by employing a higher-order quantization condition. However, in most cases of interest one must then resort to numerical techniques to evaluate the resultmg integrals. An zrlternstc spproach 1sthe incorporation of the L.anger [4,5] correction for radrsl potenrials rnto the first-order q~nt~tion condition. The purpose of rhis paper IS to report a
tractive for the stratghtforw~rd evaluation of reliable hnetic 3nd potential energy exprctatton wtues.
energy can be 369
Volume 87, number 4
2 April I982
CHEMICAL PHYSKS LETTERS
2. Theory
We begin wtrh the modtkd
quantizatlon eonditton
(u + ;) = (Il’x)jkR,
(9)
where P’= {?/A[&- U(R) - AV(R)]}1’2,
=-(l/27r)$(l/2R2P)dR +(I/%)$!i
[~AV(R)j2R’P3]dR
WI
and &V(R) is rhe Langer [4] correctIon given by
the expression [6]
= -lae
+ ~)~~~]~/~a(~~ f)laq,,
(17)
and eq. (15). Weobtain the result (T),,, = #“,
08)
whereB, is a rotationalconstantin the famhar Ed pression
+H,[J(J+ I)J3 +...
(191
for the ~bratiooal-rotational energy levels of a dtatomic molecule. Justification for the approxtmations made in eqs. (15b) and (16b) is given in the appendix.
3. Diicussion In section 2 it was shown that the sernicl~si~~ expectation value for the kinetic energy of 3 diatomic moIecule can be written as (13 = (T), f
W)
where(T), E the frst-order result given in eq. (8). and W&r is a correction ensuing from the incorporation of Langer’s (41 modification into the first-order quotation condition. The ~it~g beha~or of’ W1 has already been discussedby Stwalley [I] and
we immedfately recognize rhe first term on the right. of eq. (14) as Stwrilley’s formula for (T). The second term ISa correction Wmrr which is easily evaluated from
370
need not be repeated here. The limiting behavior of mm1 ISst~ghtfo~ard to derive. Co~lde~g vibrational energy levels only, we write the low u Iimit of (%orr in terms oi the Dunham [7] coefficients
Vofume 87, number 4
CHEhflC~L
Table 1 Cvpectatmn values of the kinetic energy for viint~ond of the X’ Zp’state of Hz (in cm-*) Ref. [l]
”
Ref. [3]
l-h~s work,
2 Apnl 1982
PHYSICS LETTERS
Appendix states
Using the notation
Exact 121
eq. (20) 0 I
2 3 4 5 6 : 9
1070
10785
1078
1079
3032 4763 6169
3040 4770 6275
3039 4770
3MO 4770
7550 8604 9424 10290 9993 10277
10
9902
11 12 13 14
9097 7757 5706 2788
7561 8630 9485 10543 10123 10740 10707 10137 9918 9139 8084
6275 7556 8610 9429 10294 9998
6275 7555 8609 9327 10292 9996
1028)
10278
9906 9100 7760 5708 2789
9904 9098 7754 5713 2755
and the first-order quantr~tion condition, Le Roy f6j expresses the rotational constant B, in the form
and the centnfugal
distortion constant D, as
=D(')+D(')+D(')
D ”
u
”
”
3
where
Da) = (I /8~‘,p0:$4:5/(48)”
Ok31 = -(l/6~‘)$$(po:‘;l)2/(4j@3, WC IlOW
consider eq. Ii%), which cm be rewritten
in
the form while near the dlssocration
limit (T&,,,
can be ex-
[a(u+i)/aE],
= (1/2rr)/$;(l
+a),
pressed as [6] (%x,
= ;xl(ll)(u,
Zn/(n- a)- 2 - u)
where
(22)
where n is the leading power in the long-range eapmsion for the potential
D IS the dissociation energy, uD is the effective vlbrational quantum number at dissociation, and X, is a constant which depends on P, n, and C, [?I. Nurnc~c~ results for the X 1Zi state of Hz are listed in table I as an example of the improved accuracy obtamed when 02Wrr IS added to the kinetic energy expectation values reported by Stwalley [I ]_ Also listed for comparison are the values calculated by Tipping [3]. The low IJexpression for (ZJcorr was used for d levels since the t$, value is not known vvlth czaainty, and the applicabihty of eq. (22) for this state is questionable fl]. The necessary Y,] constants, valid for levels up to u = 8, were taken from the recent compilation by Huber and Herzberg [S].
Smce, generally, B, 1slarger than D, by severai orders of magnitude it in~mediate~y follows that la 1% 1. S~mtiarly. we rewrite eq. (16s) as
[a+ + ;)/ax], = -(1/37)@(1-B). tihere
from wiuch we conc!ude that &I+ 1 holds, even for systems with a large reduced mass.
References [I] WC. Stw~iicy, J. Chcm. Phyr 58 (1973) 3867. [2] R.J. Lc Roy and R.B. Bernnein. J Chem. Phys. 49 (1968) 4312. [3] R.H.T~ppmng, J. hfol Spwtry. 52 (1974) 177. [? J RX. Langer. Phys. Rev. 5 1 (1937) 669.
371
Volume 87, numbcr
1
CHEMKAL
[5 1 C. Roscnzacg and J.B. Rriegcr, J. Math. Phys. 9 (1968) 849, J 8. Krqer. J. hl~h Phys 10 (1969) 1W; J.E. Adams and \V H. hldlrr. J. Chcm. Phys 67 (1977) 5775, rnd rcfcrcnccs thrrcm.
372
PHYSICS LETTERS
2 April 1982
[61 RJ. Le Roy, in: Scmicbsslal mcfhods m moleculilc sc~enng and spectroscopy, ed. MS. Child (Reid& Dordrccht, 1980) pp. 109-126. [7] I L. Dunhrm, Phys. Rev. 41(1932) 713,721. [S] K P Nuber and G. Hcnberg.Constnntsofd~tomic molecules (WUINostrand. Princeton. 1979).
CHEMICAL PHYSICS LE-JTERS
Votume 87. number 4
IMPROVED SEMICLASSICALCALCULATION OF ENERGY EXPECTATION VALUES FOR DIATOMIC MOLECULES PA. VICHARELL~
and C.B. COLLINS
Recewed 28 Scptcmber 1981; in tinal form 1.1Januxy 1982
Scmichsslwl e\prewonr for fhc c\pecr.Mon VJIW of the kinctrc and potcntul cncr&ys of ;1 dlJtOmlc malcculc 311: denvcd using 3 modrficd quanruation condrrlon Apphatron of this trcxmcm to rhc 1Xi state oi 111results m Improwti numcr~uI3grc~mcnt
wtrf,
the accurate
qo3nIal e\pccratlon
values
written in the form
I. Introduction The ctasstcal expectation
value of the kinetic energy of a dialonlic molecule IS given m atomic units by
rhe ehpressmn, (T) 3 $kdr/$dl
0)
= ~~~~I~~~P)~,
(3
tr>=4(u+~)dEld(u+f)=IdEldIn(u+~).
(8)
This expression compared very well with accurate
results [2] for H2, although small drfferences could be observed. Following a different treatment, Tippmg [3] obtained more precise expressions for (TI and (19 m the low ulimtt, by using the D~ham oscrktor 3pproxlmation His results were excellent uptolJ=3.
where P= {2p[E-
U(R)])“_‘,
(3)
U(R) = V(R) + X&R’,
(4)
and X=J(J+
1).
6)
Here T, V, and E are the kinetic, potenrial, and total energies, respecttvely, and ~1is the nuckar reduced IIIWS. The expectation value of the potential energy is obtained from
(13=E-
Ul.
(6)
the especratron vah~eof the
(7) kinetic
simple anslytical result obtained from the latter
present deriv3tion differs from earher the modified quantiz3tion condition takesinto account the f3ct thst the potential is defined on th‘e (0, m) semi-infirute interval, and not on mAod.
work
The
in that
the (--,
-) infirute
interval
requwec!
for cq. (7). As
expected, the improved equations dimimsh the prcvlousiy observed g3p between semrclassica1and quantal results, and their suttpkity makes them very at-
Stwalley [l] has shown that by using the firstorder semiclassical quantization condition (u f I) = (I/~~)~~~,
In prmciple, more accurate results may be obtsined for all v by employing a higher-order quantization condition. However, in most cases of interest one must then resort to numerical techniques to evaluate the resultmg integrals. An zrlternstc spproach 1sthe incorporation of the L.anger [4,5] correction for radrsl potenrials rnto the first-order q~nt~tion condition. The purpose of rhis paper IS to report a
tractive for the stratghtforw~rd evaluation of reliable hnetic 3nd potential energy exprctatton wtues.
energy can be 369
Volume 87, number 4
2 April I982
CHEMICAL PHYSKS LETTERS
2. Theory
We begin wtrh the modtkd
quantizatlon eonditton
(u + ;) = (Il’x)jkR,
(9)
where P’= {?/A[&- U(R) - AV(R)]}1’2,
=-(l/27r)$(l/2R2P)dR +(I/%)$!i
[~AV(R)j2R’P3]dR
WI
and &V(R) is rhe Langer [4] correctIon given by
the expression [6]
= -lae
+ ~)~~~]~/~a(~~ f)laq,,
(17)
and eq. (15). Weobtain the result (T),,, = #“,
08)
whereB, is a rotationalconstantin the famhar Ed pression
+H,[J(J+ I)J3 +...
(191
for the ~bratiooal-rotational energy levels of a dtatomic molecule. Justification for the approxtmations made in eqs. (15b) and (16b) is given in the appendix.
3. Diicussion In section 2 it was shown that the sernicl~si~~ expectation value for the kinetic energy of 3 diatomic moIecule can be written as (13 = (T), f
W)
where(T), E the frst-order result given in eq. (8). and W&r is a correction ensuing from the incorporation of Langer’s (41 modification into the first-order quotation condition. The ~it~g beha~or of’ W1 has already been discussedby Stwalley [I] and
we immedfately recognize rhe first term on the right. of eq. (14) as Stwrilley’s formula for (T). The second term ISa correction Wmrr which is easily evaluated from
370
need not be repeated here. The limiting behavior of mm1 ISst~ghtfo~ard to derive. Co~lde~g vibrational energy levels only, we write the low u Iimit of (%orr in terms oi the Dunham [7] coefficients
Vofume 87, number 4
CHEhflC~L
Table 1 Cvpectatmn values of the kinetic energy for viint~ond of the X’ Zp’state of Hz (in cm-*) Ref. [l]
”
Ref. [3]
l-h~s work,
2 Apnl 1982
PHYSICS LETTERS
Appendix states
Using the notation
Exact 121
eq. (20) 0 I
2 3 4 5 6 : 9
1070
10785
1078
1079
3032 4763 6169
3040 4770 6275
3039 4770
3MO 4770
7550 8604 9424 10290 9993 10277
10
9902
11 12 13 14
9097 7757 5706 2788
7561 8630 9485 10543 10123 10740 10707 10137 9918 9139 8084
6275 7556 8610 9429 10294 9998
6275 7555 8609 9327 10292 9996
1028)
10278
9906 9100 7760 5708 2789
9904 9098 7754 5713 2755
and the first-order quantr~tion condition, Le Roy f6j expresses the rotational constant B, in the form
and the centnfugal
distortion constant D, as
=D(')+D(')+D(')
D ”
u
”
”
3
where
Da) = (I /8~‘,p0:$4:5/(48)”
Ok31 = -(l/6~‘)$$(po:‘;l)2/(4j@3, WC IlOW
consider eq. Ii%), which cm be rewritten
in
the form while near the dlssocration
limit (T&,,,
can be ex-
[a(u+i)/aE],
= (1/2rr)/$;(l
+a),
pressed as [6] (%x,
= ;xl(ll)(u,
Zn/(n- a)- 2 - u)
where
(22)
where n is the leading power in the long-range eapmsion for the potential
D IS the dissociation energy, uD is the effective vlbrational quantum number at dissociation, and X, is a constant which depends on P, n, and C, [?I. Nurnc~c~ results for the X 1Zi state of Hz are listed in table I as an example of the improved accuracy obtamed when 02Wrr IS added to the kinetic energy expectation values reported by Stwalley [I ]_ Also listed for comparison are the values calculated by Tipping [3]. The low IJexpression for (ZJcorr was used for d levels since the t$, value is not known vvlth czaainty, and the applicabihty of eq. (22) for this state is questionable fl]. The necessary Y,] constants, valid for levels up to u = 8, were taken from the recent compilation by Huber and Herzberg [S].
Smce, generally, B, 1slarger than D, by severai orders of magnitude it in~mediate~y follows that la 1% 1. S~mtiarly. we rewrite eq. (16s) as
[a+ + ;)/ax], = -(1/37)@(1-B). tihere
from wiuch we conc!ude that &I+ 1 holds, even for systems with a large reduced mass.
References [I] WC. Stw~iicy, J. Chcm. Phyr 58 (1973) 3867. [2] R.J. Lc Roy and R.B. Bernnein. J Chem. Phys. 49 (1968) 4312. [3] R.H.T~ppmng, J. hfol Spwtry. 52 (1974) 177. [? J RX. Langer. Phys. Rev. 5 1 (1937) 669.
371
Volume 87, numbcr
1
CHEMKAL
[5 1 C. Roscnzacg and J.B. Rriegcr, J. Math. Phys. 9 (1968) 849, J 8. Krqer. J. hl~h Phys 10 (1969) 1W; J.E. Adams and \V H. hldlrr. J. Chcm. Phys 67 (1977) 5775, rnd rcfcrcnccs thrrcm.
372
PHYSICS LETTERS
2 April 1982
[61 RJ. Le Roy, in: Scmicbsslal mcfhods m moleculilc sc~enng and spectroscopy, ed. MS. Child (Reid& Dordrccht, 1980) pp. 109-126. [7] I L. Dunhrm, Phys. Rev. 41(1932) 713,721. [S] K P Nuber and G. Hcnberg.Constnntsofd~tomic molecules (WUINostrand. Princeton. 1979).