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Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

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646	miljenko peric´ and sigrid d. peyerimhoff

The zeroth-order vibronic level is threefold degenerate, with the wave functions ju u K uþ1 K u þj i j1i; ju u K uþ1 K u 1 þi j2i and ju u 2 K uþ1 K uþ1þi j3i. The zeroth-order energy is given by Eq. (A.18). The ﬁrst-order energy corrections are

E2ð1Þ ¼ Aso E1ð1=Þ3 ¼ D

where

D ¼ 2A2so þ u2 þ t2

The corresponding wave functions are

Aso

2 ¼ pu2

þ t2 j2i pu2 þ t2 j3i

rj i

2 rj

1=3 ¼ p

Aso

1 D 2

p p

rj i

			þ
The second-order energy correction are
E2ð2Þ	¼	1

		8ðuT eT2 oT2 þ uCeC2 oC2 Þ

fuT ðuT þ 1Þ½ðuT þ 2Þe2T o2T þ uCe2Co2C&e2T oT þ uCðuC þ 1Þ½uT e2T o2T þ ðuC þ 2Þe2Co2C&e2CoCg

ðA:22Þ

ðA:23Þ

ðA:24Þ

ðA:25Þ

E1ð2=Þ3 ¼

ðuT þ 1Þ 1 þ

eT2 oT

ðuC þ 1Þ 1

eC2 oC

u2 þ t2

Aso

ðuT þ 1Þ

eT2 oT þ ðuC þ 1Þ

eC2 oC

u2 þ t2

ðA:26Þ

The formulas for E1=3 are valid also for the case uC ¼ 0: They are identical to (A.17) if in the latter, K is replaced by uT 1.

The Aso ¼ 0 limits of the formulas presented in this subsection cover all particular cases (including uT ¼ 1; uC ¼ 1, and uT ¼ 2; uC ¼ 1) handled in the previous works [18,132,153].

renner–teller effect and spin–orbit coupling

647

APPENDIX B: PERTURBATIVE HANDLING OF THE RENNER–TELLER EFFECT AND SPIN–ORBIT COUPLING INELECTRONIC STATES OF TETRAATOMIC MOLECULES

We restrict ourselves again to symmetric tetraatomic molecules (ABBA) with linear equilibrium geometry. After integrating over electronic spatial and spin coordinates we obtain for electronic states in the lowest order (quartic) approximation the effective model Hamiltonian H ¼ H0 þ H0, which zerothorder part is given by Eq. (A.4) and the perturbative part of it of the form

¼ aT oT qT4 þ aCoCqC4

þ b0poT oCqT2 qC2

T oT

2i fT

CoC

o o

q2 2

2i fT

4ifT

e4ifT

& þ

e 4ifC

þ e4ifC & þ cTCpoT oCqT2 qC2 ½e 2ifT e 2ifC

e2ifT e2ifC

2 Aso

B:1

The dimensionless parameters aT ; . . . ; cTC appearing in the last expression are connected with the sums and differences of the adiabatic potentials as shown elsewhere [149,150]. This effective Hamiltonian acts onto the basis functions (A.1) with ¼ 2.

The zeroth-order Hamiltonian and the spin–orbit part of the perturbation are diagonal with respect to the quantum numbers K; ; P; uT ; lT ; uC, and lC. The terms of H0 involving the parameters aT ; aC, and b0 are diagonal with respect to both the lT and lC quantum numbers, while the b2 term connects with one another the basis functions with l0T ¼ lT 2; l0C ¼ lC 2. The c terms couple with each other the electronic species and . The selection rules for the vibrational quantum numbers are u0T=C ¼ uT=C; uT=C 2; uT=C 4.

As in the case of electronic states of tetraatomic molecules, because of generally high degeneracy of zeroth-order vibronic leves only several particular (but important) coupling cases can be handled efﬁciently in the framework of the perturbation theory. We consider the following particular cases:

Case Ba	uC ¼ 0
Case Ba1	K ¼ uT þ 2
The zeroth-order vibronic wave function is juT uT		0 0 þi. The zeroth-order
energy is
	Eð0Þ ¼ ðuT þ 1Þ oT þ oC	ðB:2Þ

648

miljenko peric´

and sigrid d. peyerimhoff

The ﬁrstand second-order energy corrections are

¼ ðu1

Þðu

2þ ðu þ

2 Þ

2 Aso

Eð1Þ

2 aT T

2aC C

1 b0poT oC

Eð Þ ¼

ðuT þ 1ÞðuT þ 2Þð4uT þ 9ÞaT oT 9aCoC

ðuT þ 1ÞoT þ oC þ

oT oC

ðuT þ 1Þb0

oT þ oC

oT oC

b2 4uT oT þ ½ðuT þ 1ÞðuT þ 2Þ þ

þ uT ðuT 1Þ

oT þ oC

oC oT

1ð

Þð

ð 2

2 uT

2 aT b0poT oC

4 uT

1 aCb0poT oC

B:3

ðuT þ 1ÞðuT þ 2ÞðuT þ 3ÞðuT þ 4ÞcT oT 6cCoC

oT oC

ðuT þ 1ÞðuT þ 2ÞcTC

oT þ oC

Case Ba2

K ¼ uT

The zeroth-order wave function is juT

uT 2 0 0 þi

þ 2 Aso

Eð1Þ

uT ðuT

þ 5ÞaT oT þ 2aCoC þ ðuT þ 1Þb0poT oC

Eð Þ ¼

uT ðuT þ 1Þð4uT þ 35ÞaT oT 9aCoC

þ oC þ ð

oC oT

2 0 ðuT þ

oT þ ðuT þ

ÞoC

þ oT

2uT oT oC

1ÞoT oC

oT oC

b2 ð12uT 8ÞoT þ ðuT

þ uT

6Þ

ðB:4Þ

oT þ oC

þ ðuT 1ÞðuT

2Þ

oT oC

oC oT

2þ

ðu

2uT

aT b0poT oC

aCb0poT oC

uT ðuT þ 1ÞðuT þ 2ÞðuT þ 35ÞcT oT 6cCoC

oT oC

cTC 4uT oT

þ uT ðuT þ 1Þ

oC þ oT

Case Ba3 K < uT

renner–teller effect and spin–orbit coupling

649

The zeroth-order energy level is twofold degenerate. The corresponding vibronic basis functions are juT K þ2 0 0 i j1i and juT K 2 0 0 þi j2i. The ﬁrst-order energy correction is

1=2 ¼

T þ

þ ðu

Eð1Þ

3u2

K2 aT

2aCoC

b0poT oC

B:5

where

D ¼

16ðKaT oT þ AsoÞ2 þ 9ðuT2 K2Þ½ðuT þ 2Þ2 K2&cT2 oT2

B:6

The corresponding vibronic wave functions are of the form (A.13) and

(A.14) with D given by (B.6) and

B ¼ 4ðKaT oT þ AsoÞ

ðB:7Þ

The second-order energy corrections are of the form

E1ð2Þ ¼ c112 H11 þ c122 H22 þ 2c11c12H12

ðB:8Þ

E2ð2Þ ¼ c212 H11 þ c222 H22 þ 2c21c22H12

where

H11=22 ¼

ðuT

þ 1Þ½17uT

ðuT þ 2Þ 9KðK 4Þ&aT2 oT 9aC2 oC

b02 4ðuT

þ 1Þ2oT þ 4ðuT þ 1ÞoC

oT oC

þ ½uT ðK 2Þ &

þ ðuT

KÞðuT k þ 4Þ

oC oT

oC þ oT

b22 4½uT ðuT þ 2Þ KðK 4Þ 4&oT þ ½uT ðuT

2Þ

þ KðK 4Þ þ 4&

oT oC

þ ½uT ðuT þ 6Þ þ KðK

4Þ þ

12&

oT oC

oC oT

oC þ oT

1½ u ð

ð 2

Þ&

3 T

K K

4 aT b0poT oC

1 aCb0poT oC

ðuT þ 1Þð2 KÞcT oT ½17uT ðuT þ 2Þ 3Kð5K 12Þ& 6cCoC

1 2

ÞðuT K þ 2ÞoT þ ðuT KÞðuT

oT oC

cTCf4ðuT K

K þ 2Þ

oC oT

þ ðuT KÞðuT K þ 2Þ

oT oC

ðB:9Þ

oC þ oT

650

miljenko peric´ and sigrid d. peyerimhoff

and

H12 ¼

ðuT2 K2Þ½ðuT þ 2Þ2 K2&

ðuT þ 1ÞaT cT oT

ðB:10Þ

þ 3b0cT poT oC þ

2 b2cTC 4oT þ oC

þ oC

oT oC

If we put oT ¼ o; oC ¼ 0; aT ¼ a; cT ¼ c; aC ¼ b0 ¼ b2 ¼ cC ¼ cTC ¼ 0 all the formulas for case (Ba) reduce to those describing the R–T effect in triatomic molecules.

Case Bb K ¼ uT þ uC þ 2ð¼ KmaxÞ

The zeroth-order vibronic wave function is juT uT uC uC þi. The zerothorder energy is

Eð0Þ ¼ ðuT þ 1ÞoT þ ðuC þ 2ÞoC

ðB:11Þ

The ﬁrstand second-order energy corrections are

Eð1Þ ¼ ðuT þ 1ÞðuT þ 2ÞaT oT þ ðuC þ 1ÞðuC þ 2ÞaCoC

þ ðuT þ 1ÞðuC þ

þ 2 Aso

ðB:12Þ

1Þb0poT oC

Eð2Þ ¼

ðuT þ 1ÞðuT þ 2Þð4uT þ 9ÞaT2 oT

ðuC

þ 1ÞðuC þ 2Þð4uC þ 9ÞaC2 oC

b0ðuT

þ 1ÞðuC þ 1Þ ðuT þ

1ÞoT þ ðuC þ 1ÞoC þ

oT oC

oC þ oT

b22 2uT ðuC þ 1ÞðuC þ 2ÞoT þ 2uCðuT þ 1ÞðuT þ 2ÞoC

þ ðuT uCÞð2uT uC þ uT þ uC 1Þ

oT oC

oC oT

oT oC

þ ½ðuT þ

1ÞðuT þ 2Þ þ ðuC þ 1ÞðuC þ 2Þ&

oC þ oT

2ðuT þ

1ÞðuT þ 2ÞðuC þ 1ÞaT b0poT oC

1ð

Þð

þ Þ

1 uC

2 aCb0poT oC

ðuT þ 1ÞðuT þ 2ÞðuT þ 3ÞðuT þ 4ÞcT oT

ðuC þ 1ÞðuC þ 2ÞðuC þ 3ÞðuC þ 4ÞcC2 oC

oT oC

ðuT þ 1ÞðuT þ 2ÞðuC þ 1ÞðuC þ 2ÞcTC

ðB:13Þ

oC þ oT

renner–teller effect and spin–orbit coupling

651

For uC ¼ 0, the formulas (B.12) and (B.13) reduce to (B.3).

Case Bc K ¼ uT þ uCð¼ Kmax 2ÞuC > 0

The zeroth-order level is twofold degenerate. The corresponding vibronic basis functions are juT uT uC uC 2 þi j1i and juT uT 2uC uC þi j2i. The zeroth-order energy is (B.11). The ﬁrst-order energy correction is

E1ð1=Þ2 ¼ ðuT2 þ 4uT þ 1ÞaT oT þ ðuC2 þ 4uC þ 1ÞaCoC

þ ð

Þð

2 Aso

B:14

b0poT oC

1 D

where

D ¼ 2 ½ðuT 1ÞaT oT ðuC 1ÞaCoC&2 þ 16uT uCb22oT oC

B:15

The corresponding wave functions have the form (A.13) and (A.14). In the present case, D is given by (B.15) and

B ¼ 2ðuT 1ÞaT oT þ 2ðuC 1ÞaCoC

ðB:16Þ

The second-order energy corrections have the form (B.8) with

H11 ¼

ðuT þ 1ÞðuT

þ 2Þð4uT þ 9ÞaT2 oT

uCðuC þ 1Þð4uC þ 35ÞaC2 oC

þ 1Þ ðuT þ 1ÞðuC þ 1ÞoT

oT oC

b0ðuT

þ ðuC þ 1Þ

oC ðuC 1Þ

oC oT

oT oC

2uC

oC þ oT

b2 2ðuT uC 2uT þ 7uT uC 6uT þ 6uC

4ÞoC

oT oC

þ 2uT uCðuC þ 1ÞoT þ ð3uT

þ uC þ

9uT þ uC þ 6Þ

oC þ oT

oT oC

þ ð2uT uC

2uT uC

uC þ 4uT uC 3uT þ

3uC 2Þ

oC oT

1ð

Þð

1 aT b0poT oC

2uC

aCb0poT oC

4 ðuT þ 1ÞðuT þ 2ÞðuT þ 3ÞðuT þ 4ÞcT oT 4 uCðuC þ 1ÞðuC þ 2ÞðuC þ 35ÞcCoC

1 2		oT oC
	cTCuCðuT þ 1ÞðuT þ 2Þ 4oC þ ðuC þ 1Þ		ðB:17Þ
2		oC þ oT

652		miljenko peric´				and sigrid d. peyerimhoff
and

H12 ¼ puT uCf12ðuT þ 1ÞaT b2poT oC þ 12ðuC þ 1ÞaCb2poT oC
þ b0b2 4ðuT þ				1ÞoT þ 4ðuC þ 1ÞoC þ ðuT þ uC þ 2Þ				oT oC
								oC þ oT
	4 uC	1 uC		2 cCcTCpoT oC					B:18
			oT oC				þ 2ÞcT cTCpoT oC
þ ðuT uCÞ			oC	oT	þ 4ðuT þ 1ÞðuT
þ ð þ		Þð		þ Þ		g	ð		Þ

The expression for H22 is obtained by interchanging indexes T and C on the right-hand side of Eq. (B.17) (cCT cTC). For uC ¼ 0; E2ð2Þ ¼ H22 and the second-order energy formula for E2 reduces to that derived for the case Ba2.

Case Bd uT ¼ 1; uC ¼ 1; K ¼ 0

In other cases, the zeroth-order vibronic levels are generally more than twofold degenerate and the perturbative handling is much more complicated. An exception is the case uT ¼ 1; uC ¼ 1; K ¼ 0 with the twofold degenerate zeroth-order level. The basis functions are j1 1 1 1 i j1i and j1 1 1 1 þi j2i. The zeroth-order energy is


	Eð0Þ ¼ 2oT þ 2oC						ðB:19Þ
The ﬁrst-order energy correction is
1=2 ¼	o þ	o þ			2		ð		Þ
Eð1Þ	6aT T	6aC C		4b0poT oC	1	D		B:20
Eð1Þ	6aT T	6aC C		4b0poT oC		D		B:20
where	q
	q
	D ¼ 4 2Aso2 þ 4cTC2 oT oC						ðB:21Þ
The second-order energy correction is				D H12
	E1ð2=Þ2 ¼ H11			D H12			ðB:22Þ
			8cTCpoT oC

renner–teller effect and spin–orbit coupling

653

with

oT oC

H11 ¼ 39aT oT

39aCoC

2b0 2oT þ 2oC

oC þ oT

oT oC

24aT b0poT oC 24aCb0poT oC

12b2 oT þ oC þ

oT oC

54cT oT 54cCoC

2cTC 2oT þ 2oC þ

ðB:23Þ

oC þ oT

8b0cTCoT

8b0cTCoC

4b0cTC oT o

B:24

H12 ¼ 24ðaT cTC

þ aCcTC þ 2b2cT þ 2b2cCÞpoT oC

ð Þ

oC þ oT

Acknowledgments

We wish to thank all our colleagues who collaborated with us in the calculations presented in the calculations presented in this chapter. One of us (M.P.) likes to thank the Deutsche Forschungsgemeinschaft for ﬁnancial support.

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