- Email: [email protected]
Modified perturbation-variation method for calculating spin-spin coupling constants
VO~UTE33, nmbe;
2
CHEMICAL PHYSICS LETTERS
Received 23 L&ember 1974 Revised manilscript receibad 3
February
1 June 1975
1975
Fermi contact nuclear spin-spin coupling constants are ca.lcu!ated by a perturbation-v~r~tion method. The trial function used includes a singukity at the nucleus ond n v,uiational term. Application to the .7-D molecule \vas carried out with an LC.40 MO wzvefor!ction expylded over 2 minimd Sbter basis set. Using rezsonable values of atomic orbital expcnents q the tzilculated vCues bracket the expcrimcn!al value O~J’HD.
The perturbation
theory
fbr the calculation
or’ Femi
formulared by Ramsey [l] 2_ld has been applied at various
levels
Usually,
‘Jle
into the virtual energ excited
nuclear orbital
spin-spin
coupling
wavefunctions
constants
In 2 number
was first
of c&ulations
of soptistication.
excited state: orbit& states
The follo4ng Srst-order
contact
to molecular
employed
of the ground
contib:ltmg
problem
in the perturbation state.
7%~ procedure
expansion leads
are constructed
to divergences
by excitation
in the expansion,
of electrons the hi&es;
most to the coupling (for example refs. [I-7!),
is .ko be considered:
Are these
excited
states
a p2rticularly
good
set for expanding
the
perturbed wavefunction? Giving tha_ r11rst order perturbation
When equation. for the hydrogen atom, Schwartz [8] found ‘iat, with a Ermi contact pertur’oation cperator, singular terms appear in the corresponding perturbed function and that second order perturbation energy ?p_nds towards an infinite value.
To avOid these difficulties, two possibilities may be considered. (aj Ti;.e use of a !ess sin@ Ir operator than the Ferm’contact one [l&12]. cas?cel the singularities (b) The introduction, in ‘-3;e Fertkrbsd function, of terms which exactly ii~ order perturbation equation in the molecular case. Th!s equation can Se written: (-Fi,-EO)I#
= -_J$&),
,
arising
in tit:
(11
where &x is the usua! Fermi t;oniact ozrturbation o_~,rator, kisJ = 2; 4.~~6(rjN]Siz. (Subscripts r’2nd N refer to nuclei res+tively.) !I$,$‘, is the first order perturbed function on nuckus N, i$;,> is ihe ground state
S!TG~~O~S 2nd %WJZrurXtion.
CHEMICAL
VOILLITI~ 33, number 2 uadimiroffand
i)ou&erty
f(1) = -2 VN
c i
(!j’ilv
I June 195.5
THYSICS LETTERS
[13] tried to solve eq. (I) for the molecular
+ 2 logriN
3/ari&-)Siz1&~!
case, using a function
as ~utiined
ir, (b):
+ I#.
an equation of the form of eq. (1) where the operator A-V is rep!aced complicated form. Reasoning on these terms, the following form of l$g(1) > was chosen:
+@ satisfies
by
2 new
operator
with a more
Eq. (1) now becomes:
the following
'ihe
form for eq. (3) is found:
equatior!
which
VN = -_3 C
i
permits
us
to
determine
Ip$‘b
is
then
a~72liogOUS
to
eq. (1) where the PIX, operator
is:
(l/&)(ala~~N)s,l
which rep!aces the Fermi contlrct operator. Tl-teessential part of the singularity, the de!ta function, has thus been eliminated. function was neg!ected to ease the computational difficulties involved. For the second part, I&?) can be developed on a set of orthonormal triplet kets, bation theory.
‘Ike part containing 2s
the log
is done in c!assica! pertur-
A variationa! procedure can now be used to estimate the Cz coefkienis. The Fermi-contact NhIR coupling constant JArNr is then of the form:
This expression is divided into two parts: the first w’&h ccn be c&ula:ed from the ground skate w~ve~unction, and the second, similar 15 nature to Ramsey’s formuia (I) as generalized by .&mow [5], which t&es into xcouz: or',h.Gnorna! trip!et functions.
Volume 33, nun?ber 2
Coup!ing constnnt
ZHEI\I!CAL
I June 1975
PHYSICSLETTERS
JHD (R-= 1.4 nu)
QF ‘)
1.20 1.00 49.90
IHD b,
77.15
“E &D
2, Cxlcu!nted value using our fcrm.Jla. b, Ca!cuinted vdue using .+Jmour’s formula. C) Ckkukted value of the fist pars of our formula
1.20 1.15 58.49 100.5
1.xl 1.2@ 59.92 (6 -17) c, 100.x9
1.15 : .20 54.83
!.JlO I.20 39.52
94.87
68.05
for JHD.
3. Results Calculations were performed on the H-D molecule using a simple IAX0 MC wavefunction in a minimal Slater basis set. ITpsi.)is constFJcted from the virtual orbit& of the unperturbed function. It XII be noticed that, for (L= 1.2 (optimum exponent value for the simple hydrogen Is MC?ground state wavefunction), the value ofJtlD is found to be 59.92 1yz. This resu!t is closer to ‘the experimental value of 42.94 * 0.1 Kz [le] than the one obtained with Armour’s formula under tie same conditions (106.33 Hz). We compared the sensitivity of the couphng constant to the variation of the atomic exponent CLon both IGo> ‘anti ITgN), where &‘Fapplies to the case of $~c and ciB to iTpdV>(see table 1). These studies are compared ,tith resuIts obtained from Armcur’s formula. J,D v&es arz very sensitive to 0: variations, especially for the case of 91;. The optimum CKvalue (1.2) obtained by minimizing the ground state ener,y does not necessarily lead to a good value of the wavefunction at the nuc!eus. i;or that reason, I+ and o’~ are varied Detween 1.0 and i.2. Doing this, it can be seen that the ex~rimental value of JH* is bracketed between th: corresponding calculated values using our formula. It seems zlso interesting to consider the contribution of the first past of the expression to the total value of the coupling constant. This one rep.resenis about 10% of the total value. Further studies are no\v necessary on the problem of convergence, that is to say, the effect of an increase of the size of the basis on the stabihty of the J,,, value. This work is now in FrOpSS and wi!! be the subject of a future article.
[l? [2] [3] [S]
N.F. Ramsey, Phyz. Rev. 91 (IS53) 303. t1.M. ?kConneU, J. C&n?. Phy:;. 24 (1956) 460. T. Lowe 2nd i. Sdrm, J. Ckxn. Whys 53 (1965) 31077. J.A. Pople and D.P. Snntry, Mol. Fhys. 9 (1964) 311. [S] E.A.G. Armour, .I. Ciwm. F&E. 49 (1968) 5445. [6] C. Bz~bi-s anci G. Be!thier, ‘L%eorei.Chim. Acta 14 (1969) 71. [7j J. Kowkski, !?a VzstLi and iz.11. Roos, Chem. Phys. 3 (1974) 70. [S] C. Schwartz, Ann. Phys- 6 (195?) 156. [S] W.H. de Jeu, Mol. 2hys. 20 (19’71) 573. [IO] J.D. Power snd R.&I. Piker, Chzm. Fnynys.Letters 8 (1971) 615. [ 11) i. biot and 5. Hoarau, Compt_ Rend. Acnd. Sci. (h-is) 257C (1968) 1396. [?2] J. Holrau a.d 5. Paviot, Thewret. Cf~im. Acta 35 (1974) 243; 3. Pzviot Er!d 3. Housau, fi=o~at Chim. Acta 35 (1974) 251. [!3] T. W~dintioff~nd T.j. DOG&&~, 9. iZr~rn.firs. $7 (1967) 1581. 1141 H. Bznoi: 2nd P. Pie& Corn@, Rend. Aad. Sci. (P,is) 26% (L957) 101.
2
CHEMICAL PHYSICS LETTERS
Received 23 L&ember 1974 Revised manilscript receibad 3
February
1 June 1975
1975
Fermi contact nuclear spin-spin coupling constants are ca.lcu!ated by a perturbation-v~r~tion method. The trial function used includes a singukity at the nucleus ond n v,uiational term. Application to the .7-D molecule \vas carried out with an LC.40 MO wzvefor!ction expylded over 2 minimd Sbter basis set. Using rezsonable values of atomic orbital expcnents q the tzilculated vCues bracket the expcrimcn!al value O~J’HD.
The perturbation
theory
fbr the calculation
or’ Femi
formulared by Ramsey [l] 2_ld has been applied at various
levels
Usually,
‘Jle
into the virtual energ excited
nuclear orbital
spin-spin
coupling
wavefunctions
constants
In 2 number
was first
of c&ulations
of soptistication.
excited state: orbit& states
The follo4ng Srst-order
contact
to molecular
employed
of the ground
contib:ltmg
problem
in the perturbation state.
7%~ procedure
expansion leads
are constructed
to divergences
by excitation
in the expansion,
of electrons the hi&es;
most to the coupling (for example refs. [I-7!),
is .ko be considered:
Are these
excited
states
a p2rticularly
good
set for expanding
the
perturbed wavefunction? Giving tha_ r11rst order perturbation
When equation. for the hydrogen atom, Schwartz [8] found ‘iat, with a Ermi contact pertur’oation cperator, singular terms appear in the corresponding perturbed function and that second order perturbation energy ?p_nds towards an infinite value.
To avOid these difficulties, two possibilities may be considered. (aj Ti;.e use of a !ess sin@ Ir operator than the Ferm’contact one [l&12]. cas?cel the singularities (b) The introduction, in ‘-3;e Fertkrbsd function, of terms which exactly ii~ order perturbation equation in the molecular case. Th!s equation can Se written: (-Fi,-EO)I#
= -_J$&),
,
arising
in tit:
(11
where &x is the usua! Fermi t;oniact ozrturbation o_~,rator, kisJ = 2; 4.~~6(rjN]Siz. (Subscripts r’2nd N refer to nuclei res+tively.) !I$,$‘, is the first order perturbed function on nuckus N, i$;,> is ihe ground state
S!TG~~O~S 2nd %WJZrurXtion.
CHEMICAL
VOILLITI~ 33, number 2 uadimiroffand
i)ou&erty
f(1) = -2 VN
c i
(!j’ilv
I June 195.5
THYSICS LETTERS
[13] tried to solve eq. (I) for the molecular
+ 2 logriN
3/ari&-)Siz1&~!
case, using a function
as ~utiined
ir, (b):
+ I#.
an equation of the form of eq. (1) where the operator A-V is rep!aced complicated form. Reasoning on these terms, the following form of l$g(1) > was chosen:
+@ satisfies
by
2 new
operator
with a more
Eq. (1) now becomes:
the following
'ihe
form for eq. (3) is found:
equatior!
which
VN = -_3 C
i
permits
us
to
determine
Ip$‘b
is
then
a~72liogOUS
to
eq. (1) where the PIX, operator
is:
(l/&)(ala~~N)s,l
which rep!aces the Fermi contlrct operator. Tl-teessential part of the singularity, the de!ta function, has thus been eliminated. function was neg!ected to ease the computational difficulties involved. For the second part, I&?) can be developed on a set of orthonormal triplet kets, bation theory.
‘Ike part containing 2s
the log
is done in c!assica! pertur-
A variationa! procedure can now be used to estimate the Cz coefkienis. The Fermi-contact NhIR coupling constant JArNr is then of the form:
This expression is divided into two parts: the first w’&h ccn be c&ula:ed from the ground skate w~ve~unction, and the second, similar 15 nature to Ramsey’s formuia (I) as generalized by .&mow [5], which t&es into xcouz: or',h.Gnorna! trip!et functions.
Volume 33, nun?ber 2
Coup!ing constnnt
ZHEI\I!CAL
I June 1975
PHYSICSLETTERS
JHD (R-= 1.4 nu)
QF ‘)
1.20 1.00 49.90
IHD b,
77.15
“E &D
2, Cxlcu!nted value using our fcrm.Jla. b, Ca!cuinted vdue using .+Jmour’s formula. C) Ckkukted value of the fist pars of our formula
1.20 1.15 58.49 100.5
1.xl 1.2@ 59.92 (6 -17) c, 100.x9
1.15 : .20 54.83
!.JlO I.20 39.52
94.87
68.05
for JHD.
3. Results Calculations were performed on the H-D molecule using a simple IAX0 MC wavefunction in a minimal Slater basis set. ITpsi.)is constFJcted from the virtual orbit& of the unperturbed function. It XII be noticed that, for (L= 1.2 (optimum exponent value for the simple hydrogen Is MC?ground state wavefunction), the value ofJtlD is found to be 59.92 1yz. This resu!t is closer to ‘the experimental value of 42.94 * 0.1 Kz [le] than the one obtained with Armour’s formula under tie same conditions (106.33 Hz). We compared the sensitivity of the couphng constant to the variation of the atomic exponent CLon both IGo> ‘anti ITgN), where &‘Fapplies to the case of $~c and ciB to iTpdV>(see table 1). These studies are compared ,tith resuIts obtained from Armcur’s formula. J,D v&es arz very sensitive to 0: variations, especially for the case of 91;. The optimum CKvalue (1.2) obtained by minimizing the ground state ener,y does not necessarily lead to a good value of the wavefunction at the nuc!eus. i;or that reason, I+ and o’~ are varied Detween 1.0 and i.2. Doing this, it can be seen that the ex~rimental value of JH* is bracketed between th: corresponding calculated values using our formula. It seems zlso interesting to consider the contribution of the first past of the expression to the total value of the coupling constant. This one rep.resenis about 10% of the total value. Further studies are no\v necessary on the problem of convergence, that is to say, the effect of an increase of the size of the basis on the stabihty of the J,,, value. This work is now in FrOpSS and wi!! be the subject of a future article.
[l? [2] [3] [S]
N.F. Ramsey, Phyz. Rev. 91 (IS53) 303. t1.M. ?kConneU, J. C&n?. Phy:;. 24 (1956) 460. T. Lowe 2nd i. Sdrm, J. Ckxn. Whys 53 (1965) 31077. J.A. Pople and D.P. Snntry, Mol. Fhys. 9 (1964) 311. [S] E.A.G. Armour, .I. Ciwm. F&E. 49 (1968) 5445. [6] C. Bz~bi-s anci G. Be!thier, ‘L%eorei.Chim. Acta 14 (1969) 71. [7j J. Kowkski, !?a VzstLi and iz.11. Roos, Chem. Phys. 3 (1974) 70. [S] C. Schwartz, Ann. Phys- 6 (195?) 156. [S] W.H. de Jeu, Mol. 2hys. 20 (19’71) 573. [IO] J.D. Power snd R.&I. Piker, Chzm. Fnynys.Letters 8 (1971) 615. [ 11) i. biot and 5. Hoarau, Compt_ Rend. Acnd. Sci. (h-is) 257C (1968) 1396. [?2] J. Holrau a.d 5. Paviot, Thewret. Cf~im. Acta 35 (1974) 243; 3. Pzviot Er!d 3. Housau, fi=o~at Chim. Acta 35 (1974) 251. [!3] T. W~dintioff~nd T.j. DOG&&~, 9. iZr~rn.firs. $7 (1967) 1581. 1141 H. Bznoi: 2nd P. Pie& Corn@, Rend. Aad. Sci. (P,is) 26% (L957) 101.