Perturbation and Moment Calculations

C H A P T E R 7

Perturbation and Moment Calculations

1. Perturbation Theory A. Perturbation Expansions

T h e calculations carried out in the preceding chapter were based upon exact determinations of the eigenvalues and eigenvectors of the hamiltonian operator. There are, however, many spin systems whose magnetic resonance spectra can be analyzed by approximate calculations based on perturbation theory (1). Perturbation calculations can also be used to provide a set of approximate chemical shifts and coupling constants for use as initial parameters in an iterative analysis.

T h e perturbation method assumes that the stationary hamiltonian operator

^ = -ί2ω

/

+

' G G < G ' *

can be written in the form

j e = J ^ ^ + V, (1.2)

where ^{( 0)} is the so-called zero-order hamiltonian operator, whose eigenvalues {Ω^{7.⁰⁾} and eigenvectors {u^(r0)} are known or exactly calculable, and F i s a small correction to ^{( 0)} called the perturbation operator. When the unperturbed eigenvalues are nondegenerate, the rth eigenvector and its associated eigenvalue are given by the expansions (2)

ur = ™ u+ u ™ + u ™ + - 9 (1.3)

Qr = + Ω™ + Ω^ + fl(3> + - , (1.4)

259

260

7. PERTURBATION AND MOMENT CALCULATIONS

^ Δ

£1 „<ο)

,Λ2)

_ V V

* r.s

,y(0) _ V

(0)

fc?" S'-

^ υ = (1-7)

ß<

⁼ Σ'^ρΑ

= Χ X

X ^

where

A

= ß<°> _ ß(o>

(

(o)

vanishing denominators are to be omitted. The superscripts appended to the w's and ß ' s refer to the "order" of the perturbation calculation (2).

In general, the calculation of Qr to the kth order requires only a knowledge of ur to order k — 1, so that a Äth-order perturbation calculation (k Φ 0) means that (1.3) is to be terminated after k terms, (1.4) after k + 1 terms.

Equations (1.3) to (1.9) represent the energy eigenvalues and the expansion coefficients of the unperturbed basis vectors mixed by the perturbation operator as power series in the ratios VpJA^rH (r Φ s).

Hence a Äth-order perturbation calculation approximates the eigenvalues and expansion coefficients by polynomials of degree k — 1 in these ratios.

From a practical point of view, the perturbation method is most useful when the series expansions converge rapidly enough to be terminated after two or three terms. T h e convergence of these series can be expected to be fairly rapid whenever the absolute values of the matrix elements of V are small compared to the absolute values of the differences of the unperturbed energies. However, when the former are comparable to the latter, the perturbation expansions may diverge, or converge so slowly as to require a calculation of very high order. It is not uncommon for the convergence to be quite rapid for some eigenvalues and eigen- vectors, whereas the expansions for other eigenvalues and eigenvectors may converge very slowly or diverge. In such instances it is necessary to

1. P E R T U R B A T I O N T H E O R Y 261 diagonalize all submatrices of the hamiltonian matrix generated by unperturbed basis vectors too strongly mixed by V to be accurately described by perturbation theory. Similar remarks apply when there is a complete or partial degeneracy of the unperturbed energies (2).

The application of the perturbation method to a particular system requires the specification of 3F^M and a determination of the zero-order eigenvalues and eigenvectors satisfying

J F ( 0 )W( 0 ) = β(ο)^Μ(θ) (Γ ^{= 1?} 2, ...). (1.11) T h e choice of J F^{( 0)} will depend upon the properties of the system under

consideration, and this section will be restricted to the two limiting cases described by the inequalities:

I/ G G 'I < I "^{G G}- I , (1-12)

I/ G G 'I > I "G G< | , (1.13)

for all G, G' = A, B, ... (G Φ G').¹ A spin system will be said to be weakly coupled or strongly coupled, accordingly as its internal shifts and coupling constants satisfy ( 1.12) or ( 1.13). This classification is introduced only for the purpose of distinguishing between the various orders of perturbation in the two limiting cases, for example, first-order perturba- tion theory—weak coupling; first-order perturbation theory—strong coupling, etc.²

1. Weakly Coupled Systems. For a weakly coupled system, the spin-spin interactions are collectively treated as the perturbation operator, so that

Jf<°> = -%œGIGzJ V = - 2 2 /G G 'IG -IG ' . (1.14)

G G < G '

T h e zero-order eigenvalue problem is trivial since the eigenvectors of are just the product kets

{I / A , wA; ·•· ; /G » * « G ; — ; m = mA + mB + ···. (1.15) T h e corresponding eigenvalues are

βΓ = - Σ « ν » ο - 0-16)

G

1 T h e application of p e r t u r b a t i o n t h e o r y to spin systems w h o s e internal chemical shifts a n d s p i n - s p i n c o u p l i n g c o n s t a n t s d o n o t c o n f o r m to ( 1.12) or ( 1.13) will b e discussed in Sections 2 a n d 3. T h e t r e a t m e n t of s y m m e t r i c a l systems will be deferred to C h a p t e r 8.

2 I n t h e literature, t h e t e r m " s t r o n g l y c o u p l e d " is often used to describe a system for w h i c h

I

I « I

I,

262

7. PERTURBATION AND MOMENT CALCULATIONS

and the nonvanishing matrix elements of the perturbation operator are

< ~ ' IG > ^mo \ " ' IG ' , mG>\ ···; m \ V | ··· IG , mG ; · · · / ( - , m^G'î "S ηΐ)

= - X X/ G G 'VG ' , (1.17)

G < G '

<···/⁰ , m^G\ ~'I^G> , m^G>\ ···; m | V | ··· 7^G , m^G — 1 ; , m^G* + 1 ; ···; m>

= -yGG>F(IG ,

-m

)F(/<r

where

F(/o , mo) = [ ( /G - mG)(I^G + m^G + \ψ\ (1.19) 2. Srongly Coupled Systems. For a strongly coupled system, J ^^{( 0)}

and F are given by

^^{( 0 )} = - j ω⁰Ι^ζ

+ Χ X

2)

' G < G ' ' G

where 3tf⁽⁰⁾ is obtained from (1.1) by setting ω^Α = ω^Β = ··· = ω⁰ . T h e symbol δ denotes a small increment in the quantity immediately following it. For example, the frequencies

δα»

T h e zero-order hamiltonian admits the following constants of the motion:

I2> IA2> IB25 · · · » I ζ » Σ Σ / G G 'IG ' IG ' · G < G '

T h e eigenvectors of the square and ζ component of the total angular momentum are also eigenvectors of I^{G 2}, G = A, B, but, excepting the special case of two groups, they are not eigenvectors of the coupling operator. T h e general solution of the zero-order eigenvalue problem requires rather lengthy calculations and will not be considered here. T h e analysis will be illustrated for the A B C system in Section l . F .

In the special case of two groups, the simultaneous eigenvectors of I2, I^{A 2}, I^{B 2}, and Iz, which will be denoted³ {| / , IA , IB , m>}, are eigenvectors of Jf<°>. For I² = I^{A 2} + I^{B 2} + 21 A · I^B , so that

jr<°> =

+ ^i/(P

3 T o simplify t h e notation, t h e spin multiplicity index s j will n o t b e explicitly indicated in t h e e l e m e n t s of t h e basis.

1. P E R T U R B A T I O N THEORY 263

4 T h e total spin q u a n t u m n u m b e r s 7 a n d 7 ' are h e r e u n d e r s t o o d to b e t a k e n from t h e series 7A + 7B , 7A + 7B — 1, | 7A — 7B |. T h e multiplicities of t h e s e q u a n t u m n u m b e r s m a y be d e d u c e d b y t h e p r o c e d u r e d e s c r i b e d in Section 3.E, C h a p t e r 4.

Hence the eigenvalues of J f^{( 0 )} are

QZAJB,m = " Κ ™ + tiW + 1) - + 1) - /„(/„ + 1)]}· (1-22) T o complete the theoretical basis for the case of two groups, it is necessary to calculate the matrix elements of I^AZ and I^BZ. Since Ugz , Q = 0, (G = A, B), and IAZ + IBZ = 4 , it follows that⁴

</',

I

Φ

<T, IA,I^Bym

I

I

\ I

\I

I

,I

, ™>

Hence

iL I κ Jb , ™ \ I*z \ 1,1 α , h > m> = m — (I,I

I

m \ I

\ I, I

, I

, w>,

(1.23)

<Λ

</',

I, I

,I

,

(1.24) Consider now the operator identity:

[Ρ, [I², /^AJ] - I⁴/^{A 2} - 2 P /^{A z}P + /^{A 2}P

= 2 { P /^{A 2} + 7^{A e}P - 7,(F - V + I^{A 2}) } , (1.25) which may be verified by a straightforward calculation based on the commutation rules for the spin operators. Computing the matrix elements of this identity that are diagonal in m but nondiagonal in I and one obtains

[(7 + / ' + l )² - 1][(7 - 7')² - 1]</', /A, / B , « | / A ,I l I ^a > h , m) = 0, by virtue of the fact that the matrix elements of 7²( P — I^{B 2} -f I^{A 2}) vanish for Γ Φ I. T h e matrix elements of I^AZ for I Φ Τ will be non- vanishing if + l )² — 1 = 0 , which implies that / + Τ = 0 or I + Γ = - 2 , or if (I — Ff - 1 = 0 , which implies that Γ =1 ± 1.

Since the spin quantum numbers are nonnegative, the only possibilities are

Γ = I± 1. (1.26)

264 7. PERTURBATION AND MOMENT CALCULATIONS

T h e matrix elements of IAz diagonal in I and m are, by (1.25), (II I m\I \II I m- - m W + l ) ~ / b ( /b + 1 } + / a ( /a + 1 )]

(1.27) W h e n /A = IB , the right side reduces to m/2, so that the matrix elements are not explicit functions o f / ; however, their number is delimited by 7.

This leads to the conclusion that the diagonal matrix elements of IAz vanish if 7 = 0. For 7 = 0 can result only by the addition of two equal spin quantum numbers: IA = IB Φ 0 (i.e., I = IA — IA = 0) or IA = IB = 0. T h e value of m is zero in either case, so the matrix elements vanish, as asserted.

T h e diagonal matrix elements of 8Z may be obtained from (1.23), (1.24), and (1.27):

<A , , m

I

I

„Λ 4- " A B [ / A ( / A + 1) - / „ ( / B + 1) ~ / ( / + 1)] \ N ? Rv

- ~ " Γ

27(Τ+Ί) S'

where ΩΑ Β = Δ ΩΑ — ΔΩΒ . These matrix elements may be used to calculate the first-order spectrum of two strongly coupled groups.

Second-order theory requires the matrix elements of IAz connecting states with Γ — I = ^ 1. T h e second-order theory is rather lengthy and seldom used, so that the derivation of these matrix elements will be omitted.5 T h e results are

< / + l , /^A, /^B, « | /^{A i}| / , /^A, /^B, « ) = C ( 7 + 1,7; 7^A , 7^B)[(7 + If - τηψ\

(1.29)

</ - 1, 7A , Λ,, « I / A * I Λ /A , / Β , ™ > = C(Î - 1, 7; 7A , 7B)(72 - πίψ\

where

C ( / - l, / ; /^A, 7^B)

= C ( / , 7 - 1 ; /A, /B)

= [(7 - 7^A + 7^B)(7 + 7^A - 7^B)(7^A + 7^B + 7 + 1)(7^A + 7^B - 7 + 1)]*/*

47²(27 - 1 ) ( 2 / + 1)

(1.30)

' T h e derivation is given in reference 3(d) of C h a p t e r 4.

1. P E R T U R B A T I O N T H E O R Y 265

B. Zero-Order Spectra

T h e zero-order hamiltonian operator for a strongly coupled system is identical to the hamiltonian operator for the A^N system (cf. Section 4.A, Chapter 5). T h e theory of the A^ system shows that the zero-order spectrum of any strongly coupled system consists of a single resonance at ω⁰ .

T h e zero-order spectrum of a weakly coupled system is also quite simple. Transitions are allowed between any two states (i.e., product kets) for which the matrix element of I~ is nonvanishing. In particular, group-G transitions are characterized by the selection rules

AIG = 0, AmG = - 1 , AmG> = 0. (1.31) From (1.16) and (1.31), it follows that the frequency of a group G

transition is c oG .

T h e relative intensity of a group-G transition is

GIG K- IQ >™ G — U ··; m — 1

I /- I

= ZIGVG + ™GWG -mG + 1), (1.32)

where gr is the multiplicity of IG . Each transition in group G leads to a resonance at œG , so that the total intensity of the resonance is

(Int)⁰ = 2"-»G £ XG^jG(I^G + m^G)(I^G - m^G + 1) = 2^n^G . (1.33)

7G

m

T h e factor

2N-nG = J J G I G. +T { 2) L1Q} G ' ^ G

introduced in (1.33), is the number of choices for all kets | IG' , mG>y, other than | IG , mG>, in the product ket | IA , m^A ; I^G , m^G ; tn).

It follows that

(Int)^A : (Int)^B : (Int)^c - = n^A : n^B : n^c : ···, (1.34)

2)(Int)^G = 2"-W. (1.35)

G

Thus the zero-order spectrum of a weakly coupled system containing η groups of magnetically equivalent nuclei consists of η resonances, one at each of the Larmor frequencies ωΑ , ω^Β , with relative intensities proportional to nA , n^B , ... .

266

7. PERTURBATION AND MOMENT CALCULATIONS

C. First-Order Spectra—Weak Coupling

In the first-order approximation, the off-diagonal elements of the hamiltonian matrix are neglected, so that the first-order energies for weak coupling are

Ω

ζ

+ ^ = -12 ^"v«o

22 ^/

^.«

^«ο'1· c

-

)

1 G G < G ' ;

T h e selection rules (1.31) are still valid and their application to (1.36) leads to the following expression for the frequency of a generic transition in group G:

ωο

+ Σ

G ' ^ G

where G' = A, B, ... Φ G.

There is one group-G resonance for each choice of the quantum numbers m^G> . Since the resonance frequencies are not explicit functions of the total spin quantum numbers, the correct number of group-G transitions is obtained by allowing each spin variable to range from its maximum to its minimum value. In particular, m^G> may range from /^G' ( m a x ) = \n^G> to —7^G'(max), so that the coupling of group G to group G' results in 2/^G'(max) + 1 = n^G' + 1 resonances in group G.

If group G is also coupled to a second group G", each of the n^G> -f- 1 resonances is split into n^G" + 1 lines, and so on. T h e coupling of group G to groups G', G", ... results in

Π

T h e relative intensity of a group-G transition is

Π

G ' ^ G ' /G

m

'

=

2*G-ifi

^Π

G ' ^ G

where

\pi^G> + m^G<J

is the degree of degeneracy of m^G> . On summing (1.38) over all m^G> Φ m^Gl one obtains (1.35).

1. P E R T U R B A T I O N THEORY 267 T o illustrate the use of (1.37) and (1.38), consider the resonances in group C of a weakly coupled system with nA = 2, nB = 1, nc = 3.

There are (nA + l)(n^B + 1) = 6 resonances in group C, corresponding to the six ways of pairing mA and mB : (m^A , m^B) = (1, ^), (0, J), (— 1, J), (1, — J), (0, — \ )y (—1, — ^). Inserting the values of nA , n^B , mA , and mB in (1.37) and (1.38), one obtains the results given in the accompanying tabulation.

MA mB I n t e n s i t y F r e q u e n c y

1 i ⁸ œc + ^{JAC + Ü B C}

0 i ¹⁶

- 1 i ⁸ ωο — / A C + £ / B C 1 - i ⁸ ωο + / A C — ^ / B C

0 - i ¹⁶

- 1 1

~ 2 8 _œc ~ JAC ~ HBC

Examples of high-resolution spectra that conform to first-order theory for weak coupling have been given in Chapter 5. T h e only remaining problem is the determination of the conditions for which the first-order approximation is valid.

A perturbation calculation of arbitrary order will be meaningful only if the series expansions for the eigenvalues and the expansion coefficients for the eigenvectors converge. For weak coupling, the perturbation expan- sions are power series in the ratios T G G ' / ^ R S > s o t n a t

^ I

I

(R Φ S), terms of degree greater than unity in these ratios can be neglected. However, these conditions relate only to matters of conver- gence. T h e validity of a first-order calculation of the resonance fre- quencies and relative intensities also requires that the corrections to these quantities predicted by a second-order calculation be unobservable.

More generally, if one performs a perturbation calculation to order η — 1, the frequency corrections ~~ <^m) must be unobservable and corrections to the line intensities of degree η — 1 in the ratios T G G ' / ^ R S

must be unobservable. Similar remarks apply to strongly coupled systems and, more generally, to any system whose theoretical spectrum is to be calculated by the perturbation method.

Consider, for example, an A2B system with | J/8 \ = 0.067. Since ρ/δ² = 0.004, it would appear, from the condition | J/8 | <^ 1 alone, that a first-order calculation would suffice. However, an examination of the fluorine magnetic resonance spectrum of C1F3 at 56.4 Mcps (Fig. 7.1) shows that the anticipated 1 : 1 doublet and 1 : 2 : 1 triplet are not observed. T h e second-order splittings, which are of the order of

268 7. PERTURBATION AND M O M E N T C A L C U L A T I O N S

Cl F, ₂

4

F I G . 7 . 1 . F l u o r i n e m a g n e t i c r e s o n a n c e s p e c t r u m of C 1 F3 at 56.4 M c p s .

J

/2S

D. First-Order Spectra—Strong Coupling

T h e first-order energies for two strongly coupled groups are, by (1.22) and (1.28),

- \mw^A + £ / [ / ( / + 1) - 7A( /A + 1) - / B ( / B + 0 ] , « ω^{Α Β}[ /^Α( /^Α + 1) - 7^B(7^B + 1) - 7(7 + 1)] |

2 / ( 7 + 1 ) (1.39)

Since first-order theory neglects the off-diagonal elements of δΖ, I² is still conserved; hence the selection rules for first-order transitions are

AU = ΔΙ* AI = As, = 0, Am 1. (1.40) It follows that the frequency of the transition \ I, IA , I^B , m}

\I,I^A,I^B,m — 1> is given by

" A B [ / A ( / A + 1) - / „ ( / B + ! ) - / ( / + ! ) ]

21(1 + 1) (ΙΦ0), (1.41)

which is independent of the spin-spin coupling constant. Furthermore, (1.41) is independent of m, so that the resonance frequencies of the 27 transitions defined by m = 7, 7 — 1, — 7 + 1 coincide. T h u s the intensity of the resonance is

X

I

I

m

= + 1 X 2 / + 1 ) } .

m

1. P E R T U R B A T I O N T H E O R Y 269 T h e application of (1.42) and (1.41) requires the calculation of 7 and gj from given values of 7A and 7B . Consider, for example, an A3B2 system, for which IA = § , ^(gx/z = 2), 7B = 1, 0. For convenience, the frequency origin will be taken at ωΒ , so that ωΑ -> ω^Α — ω^Β = δ = ωΑ Β . T h e values of 7, gj, the resonance frequencies, and their relative inten- sities are given in the accompanying tabulation. These results should be compared with the numerical data for the A3B2 system given in Appendix VI.

7B I

2 3 1 ⁵ ₂

3 2 1 ³ ₂

3 2 1 ₂ ¹

3 2 0 ³ ₂

1 2 1 ³ ₂

1

2 1 ¹ ₂

1

2 0 ¹

2

g j I n t e n s i t y F r e q u e n c y

1 35

i*

1 10

1 1

1 10 δ

2 20

**

2 2

-*»

2 2 δ

Ε. Second-Order Spectra—Weak Coupling

In second-order perturbation theory, u^r is approximated by the first two terms of (1.3), and Q^r is approximated by the first three terms of (1.4). T h e corrections to u^{r1] and Ω^{2) for weak coupling are obtained by substituting (1.17) and (1.18) in (1.5) and (1.8). T h e second-order approximations to the eigenvectors and the eigenvalues are

| 7^A, 7^B, ...,m) = I ...I^G , m^G m)

- \

2 G < G> G G ' W

- F(I^G> , m^G')F(I^G , - m^G) | ... 7^G , m^G - 1 ... 7^G> , m^Q- + 1 ...; m>}, (1.43)

^ + ^ + ^ = - X

Χ X

G G < G '

- \ X 2

2 ο<<? ωο ο '

270

T h e selection rules for allowed transition are AI^G = 0, G = A, B, and Am = — 1. It follows that the resonance frequencies and relative intensities of group-G transitions are given by

" G + Σ / O G ' % + 1 X — { W o ' + 1) - % ( % + 1) + 2m

m

'},

G ^ G 2 G ^ G WG G '

(1.45) ftoi'o - m

+ 1)(/

+ mo) j ^{1 - X} ^

1 ^Π

< G ' ^ G œGG' )G' ^ G

where all terms of degree greater than unity in the ratios /^{G G}' / w^{G G}' have been dropped from the expansion of | ( /^A , I^B , m - 1 |/~| IA , IB , m ) \2.

Summing

(1.46)

For two groups,

(1.45)

(1.46)

1

ω

+

^{1) -}

^{+ 1) + 2m}

^m

^{}, (1.47)}

S/A& B ( / A + mA)(lA - mA

+ 1) j ^{1 -} -^p-J. ^(1.48)

These results may be checked for the A^{n A}B systems by expanding the exact expressions for the resonance frequencies and relative intensities given in Chapter 6.

F. Application to the ABC System

T h e hamiltonian operator for the A B C system is

= _ |ω /ΑΑ^ _|_ coBIBz

+

+

(1.49)

Relative to the product basis {| + + + >, | + H >, | ) } , the matrix for 3TF is the direct sum of four submatrices of dimensions 1, 3, 3, 1, corresponding to the Iz eigenvalues w = f - , ^ , — -—-§-, respectively. T h e two l x l submatrices are generated by the product

1. P E R T U R B A T I O N T H E O R Y 271 kets I + + + ) and | >. These kets are eigenvectors of £F\

& \ + + + > = ß⁰l + + +>, & \ > = Ω⁰' \ >, where

Ω

^-1{ω

+

±(J

+ J

+

Ωο

= 1(

Α + ω

+

%(J

+ J

+

T h e 3 x 3 submatrix of generated by the product kets with m = + J is of the form

/Zu + Vn V12 F^{1 3} \

·**(£) = V* ?22 + ν²² V^2Z , (1.52)

\ Vn V³² Zn + Vj

where

Ζ η = ~\(ωΑ

+ ω

— a>

),

a>

),

^ 3 3 = — έ ( —^ωΑ + ω^Β + o>c),

^ 1 1 — ~ " ? ( / Α Β — JAC ~ 7BC)> ^ 2 2 = 4 ( 7 Α Β ~ TA C + Jßc)>

^ 3 3 = 4 ( / A B + / A C - / B C ) >

^ 1 2 = ^ 2 1 =

~^JBC

~^JAC

V23 = V32 = ~ Î 7 A B -

T h e matrix for J) is obtained from (1.52) by replacing Zu with Ζ a ·

T h e eigenvalues of ^ ( \ ) will be denoted Ωλ, ß² , Ω3 , (ß^ —> Ζ^ , as all 7GG' ~^ 0)> a n (i t ne corresponding eigenvectors will be denoted

« η

I + + - > +

I + - + > + «311

« 1 2

I

I

I - + +>,

« 1 3

I

I

I - + +>·

Since ^ ( \ ) is real and symmetric, it will be assumed that the matrix (a^) is orthogonal. In the limit as all JGG' ~~* 0, —>- δ^., where is the Kronecker delta.

T h e eigenvalues and the coefficient matrix for the states with m = — \ will be denoted Ω^±\ Ω², Ω³', (fl#). According to the theorem of Sec- tion 3.F, Chapter 5, the Ωί, ß / , a{j, and are related by the following identities:

ß / ( J A B > JAC y JBC) = — — JAB , — / A C > ~JBC)J

a

i.i(J

J

These identities are very useful in perturbation calculations of order > 2 .

272 7. PERTURBATION AND M O M E N T CALCULATIONS

T r a n s i t i o n " Relative intensity F r e q u e n c y

A ,

1 + + +>->

A2

1 + + ->->

+ «32(^11 + a2i)]2 ω ; - Ω1

A3 ! + - + > - * + > [«12(«21 + «3l) + «22(«ίΐ + «3l)

+ «32(«ii + <*'£ 1)]2 - Ω2

A .

1 +--> -* ^—

B i : ! + + + > - ! + --f-> [a12 + «2 2 + « 3 2 ? ^Ω2 - Ω0

B2:

1 + + -> ->

+ «s'sfall + a2l)Y ^{Ω / - Ω ,}

B3:

1 - + +>-> ^—

+ 033(αίΐ + « 2 l ) ]2 Ω[ - Ω3

B4:

1 - + ->- ^—

1 + + +> ->

c

+ «32(«Î3 + « 2 S ) ]2 Ω9' ^{- Ω2}

c

+ 032(ûl3 + « 2 3 ) ]2 Ω2 - ΩΆ

c

—

M , : l + + - > -

—

+ «3'l(«ll + «2l)]2 Ωι ^{— Ω1}

M2 : ! + - + > - > - > [«12(«22 + «32) + «22(«12 + «32>

+ «32(«12 + «22)]2 Ω2 — Ω2

M3 : ! - + + > - > - > [«13(«23 + «33) + «2s(«13 + «33)

+ «33(«13 + «23)]2 Ω3' — Ω3

α I n t h e limit as all Jqq' - > 0 .

T A B L E 7.1

RESONANCE FREQUENCIES A N D RELATIVE IN T E N S I T I E S FOR THE A B C SYSTEM

1. P E R T U R B A T I O N T H E O R Y 273 From the preceding definitions, it is not difficult to construct Table 7.1, which clearly indicates the transformation of the spectrum into itself when JQQ' -> JGG' : G^x — G4, G2 G8 , G = A, B, C; M , M<, i= 1 , 2 , 3.

1. Weak Coupling. T h e resonance frequencies and relative intensities for group A of the A B C system, as determined by second-order perturba- tion theory, are given in Table 7.2. T h e results for groups Β and C may be obtained from those for group A by cyclical permutation of A, B,

T A B L E 7 . 2

S E C O N D - O R D E R PERTURBATION A N A L Y S I S OF T H E A B C S Y S T E M FOR W E A K C O U P L I N G

F r e q u e n c y R e l a t i v e i n t e n s i t y

JAB JA C ΩΑ Β

JA B

+

ω Α Β

+

JA B JAC ΩΑ Β W A C

JA B JA C

ω A

ω^Α

JA B

+

4 ( ΩΑ Β

IS

ΩΑ Β W A C

zi

⁺

4 / ΩΑ Β

M

₊

4 ( ^ωΑ Β W A C

1 +

and C. T h e mixed transitions are not included, since their intensities are of the second degree in / Q G / ^ R S a nd > therefore, negligible in the second- order approximation. T h e first-order A B C spectrum follows from Table 7.2 by omitting all terms containing the ratios / G G / ^ G G ' ·

Table 7.2 shows that the second-order A B C spectrum is invariant with respect to a change in the sign of any coupling constant. T h e relative signs of the coupling constants begin to influence the appearance of an A B C spectrum in the third order, that is, when terms of the second degree in the ratios / G G / ^ R S m^ k e observable contributions to the resonance frequencies and relative intensities. Calculated A B C spectra in the zero-, first- and second-order approximations are shown in Fig. 7.2. In the first-order spectrum, the magnitudes of the coupling constants appear as repeated spacings in the A, B, and C quartets.

Table 7.2 shows that the same repeated spacings occur in the second- order spectrum.

274

7. PERTURBATION AND MOMENT CALCULATIONS

(α)

(b) Ι 1.1

ωΒ

HJBCH

(c)

ω0

(d) (e)

ω0

ω0

F I G . 7.2. Calculated A B C spectra: (a), (b), a n d (c) s h o w t h e zero-, first-, a n d s e c o n d - o r d e r spectra for | TGG' I ^ I ŒG G ' l> W a nd (e) s h o w t h e zero- a n d first-order spectra for

I

I ^> I

-f

An example of an A B C system that conforms to second-order theory is provided by the vinyl protons of vinyl acetate at 60 Mcps (Fig. 7.3).

An exact analysis of this system at 29.92 Mcps ( 3 ) indicates that - i^{A B} = s^BC = s^AC , where s^GG> denotes the algebraic sign of J^GG, .

2. Strong Coupling. T h e zero-order hamiltonian operator for strong coupling is

^ = -{ω0Ιζ

+

· I

+ /

I

· I

· I

},

and the perturbation operator is

δΖ = — {δω

/

+ δω

/

+

Since the square and ζ component of the total angular momentum are constants of the motion with respect to ( 0 ), the calculation of the unperturbed eigenvalues and eigenvectors is facilitated by using the

276 7. P E R T U R B A T I O N A N D M O M E N T C A L C U L A T I O N S

basis given in Table 4.6. The results of this calculation are given in Table 7.3, where

g = ( J A B - / A C ) V 3 JAB + JAC — 2 ^{JBC + ^ '}

R = [(JAB + JAC - 2 /B C)2 + 3 ( /A B - /Ac )2]1 / 2.

T A B L E 7.3

Z E R O - O R D E R EIGENVECTORS A N D EIGENVALUES OF THE A B C S Y S T E M FOR S T R O N G C O U P L I N G

E i g e n v e c t o r E i g e n v a l u e

I | , | > - έ ( 6 α >0 + / A B + JAC + JBC)

\hi> - έ ( 2 " ο + / A B + / a c + JBC)

I f , - £ > - έ ( - 2 ω0 + /A B + JAC + / B C } i f , — f > - έ ( - 6 ω0 + J A B + JAC + / B C }

d -f Q

)

1 (1 f 02)1 /2

1 (i -

f e

)

1 (1 + 0. ) i / «

1 ; 1> - Q I 1 l ; 2 » έ ( - 2 α >0 + / A B + JAC + JBC + R }

— { Q I i i ; 1> + I h h 2 » \{-2ω, + JAB - f /A C + JBC ~ R)

{ | l , - i ; 1> - Q I l , - 1 ; 2 » ± { 2 a )0 + / A B + / a c + JBC + Ä}

{ 0 I i - h 1> + I i - f , 2 » è ( 2 o >0 + JAB + JAC + / B C - R }

The zero-order strongly coupled ABC spectrum is obtained by applying the usual rules for the computation of resonance frequencies and relative intensities, together with the selection rules AI = Asj = 0, Am = — 1 . T h e theoretical spectrum is given in Table 7.4, which shows that the resonance frequency of all transitions with nonvanishing intensities is ω0 .

The first-order corrections to the energies are given in Table 7.5.

From these results it can be shown that the first-order strongly coupled ABC spectrum [Fig. 7. 2(e)] consists of a symmetrical 1 : 10 : 1 triplet.

T h e frequencies of the two satellite resonances, relative to the central resonance, are

ΐ ~ ^ 7 ) 2 | ~ y ~ (2 δ Α — δ ωω Β — 8œc) + ^~ (8ωΒ — 8œc) j.

1. P E R T U R B A T I O N T H E O R Y 277

T r a n s i t i o n Intensity F r e q u e n c y 3 2» 3 \ ι 3

2/ ^ 1 2 3 ω0

3 2> I > - i f -Έ> 4 ω0

3 2» - I> - l 3 3 \

2» 2/ 3 c u0

3 2' 3 \ _^ ι 1

i D 0 ω⁰ _{+ 2( J a b + JA C +} / b c ) + i *

3 2' i > -*1 i I; 2 ) 0 ω0 + i ( /AB + / a c + /b c ) - i * 3 2> i > - i i 0 ω0 + È( /AB + / a c + /b c ) + i * 3 2> I> - 1 I - è; 2 ) 0 ω⁰ + iÜAB + / a c + /b c ) - I * 2' | ; D1 - > l I , - I ; D 1

2' 1 2) 0 Ω0

1 2' i 2) - 1 1 , - 1 ; I) 0 ω⁰ ₊

1 2' I; 2) -1 Έ . - Έ; 2 ) 1 1 2» 3 _ 1 \

2' 2 / 0 ω0

- lu

1 2' 0 ω0 - WAE + / a c + /b c ) - I *

1 2» 0 ω⁰ _{- I( /A}B + / a c + /b c ) + I *

TABLE 7.5

FIRST-ORDER ENERGY CORRECTIONS FOR THE STRONGLY COUPLED ABC SYSTEM

State F i r s t - o r d e r energy correction

1 2» l > — \ (ΔΑ»Α + ΔΩΒ + ö a > c )

1 2» — J (ΔΩΑ + ΔΩΒ + ΔΩΣ;)

1 2' - I > J (ΔΩΑ + ΔΩΒ + ΔΑ>ο)

i l . - L > è (ΔΩΑ - f ΔΑ>Β + 8WQ)

I I . I ; D

1 + Ö2 | 6( Δ ΩΑ

- 2ΔΑ>Β — 2ΔΑ>£·) Η (ΔΑ>Β — ΔΩ^) -Q

λ / 3

I I Έ;2) Q

— 2ΔΑ>Β — 28 ω Q) (ΔΩΒ — ΔΩ<^) V 3

— 1Δ ΩΑ|

i l l2> - I D 1 ( 1 1 + 02 | 6( Δ ΩΑ

— 2ΔΩΒ — 2ÖCÜC) + (ΔΩΡ — 8WQ) -Q

λ / 3 - \Q² δ ωΑ|

I I - I; 2 )

Ι + ο Μ

V3 — 2 δ ωΑ

j

TABLE 7.4

ZERO-ORDER ABC SPECTRUM FOR STRONG COUPLING