- Email: [email protected]
Spectral density analysis of nuclear spin—spin coupling
Chemical Physics i 2 (I 976) 273-280 0 North-Holland Publishing Company
SPECTRAL
DENSITY
II. Bartree-Fock
H.B. JANSEN,
LCAO
ANALYSIS
OF NUCLEAR
studies for homonuclear
SPIN-SPIN
COUPLlNG
coupling constants*
J.A.B. LOHMAN
Received 26 hky 1975
A spectral density function has been calculated for the indirect nuclear spin-spin coupling constant Tar homonuclar coupling. The ground state wavefunction is obtained with a normal ab-initio nlculation. The sum over states approach for cnlculating the reduced coupling constant K is replaced by an integation over a spectral density function-where the integration variable is the orbital exponent of a “scanning molecular orbits’. This results in n stable method for calculating K with reasonable accuracy. The spectral density function also gives information about which excited &es give important contributions to K. lzurthermore a residual spectral density function is defined that cm be used as a test for the completeness ot’a set ofvirtunl orbit& in n sum over states calculation.
1. introduction
Accurate numerical calculations of nuclear spinspin coupling constants in molecules have proved difficult. Perhaps the first and only quantitative calculations of a spin-spin coupling constant, with an estimated error of 10% or less, are those of Meyer [I] and Kowalewski et al. [2] for HZ. Ground state wavefunctions, approaching the Hartree-Fock level, can presently be obtained for medium-sized molecules. However, the spin-spin coupling is a second-order property, involving an infinite summation over the Hilbert space of all excited electronic states of the molecule. In practice, rather few excited states may suffice if they are judiciously chosen, E.g., for H2, the single virtual MO * This work was begun when H.B.J. and P.P. were at CECAM. Bstiment 506. Universitdde Ptis XI. Orsay, France. ** Present address: Department of Physical Chemistry, Abo Akademi, Finland.
of a simple LC.40 model can give a satisfactory agreement with experiment. The main thing, then, is knowing what are these important parts of the Hilbert space. One approach to this problem is presented in ref. [3] (part I of this series) for the X-H coupling constants in hydrides XHq. There the excited state? are rcpresented in terms of an orthonormal set of continuum wavefunctions in a Huhhen screened Coulomb potential describing the heavy atom, X. In the present work we consider homonuclear coupling constants. Not having an equally neat, orthonormal and continuous basis set in the present case, we describe the excited states of the molecule with a symmetric or antisymmetric molecular orbital, consisting,of two atomic orbitals, of gaussian or Slater type, treating the orbital exponent as a continuous variable. This basis is complete, in a subspace of the Hilbert space, but it is nonorthogonal.. The relevant densities of states are discussed in section 2. The total spin-spin coupling constant is then expressed as a continuous integral over the orbital expo-
274
H.B. fanSen ef aI.lSpecml
density analysis of nuclear spin-spin
nent. The integrand provides a spectral density analysis of the coupling constant. The integration over the orbital exponent is essentially equivalent to an integration in momentum space. Our method could thus be viewed as a continuous version of the sum over states (SQS) approach. The idea of varying continuously one additional A0 orbital exponent has been used by Getzin [4], but only in order to investigate the stability of the discrete SOS. The present method is not a very accurate one, but the purpose of the work is to get a feeling for thi: shape of the spectral density function. Therefore correlation effects are not considered in this work. One purpose of this work, then, is to get a feeling for the important parts of the Hilbert space of excited electronic states. This could be helpful in assigning the deficiency of any particular virtual orbital calculation to errors in the ground state wave function or to incompleteness of the set of excited states. Secondly, while the nuclear spin-spin coupling mainly is a lowfrequency polarizabibty, one may ask whether a highfrequency tail would exist like For the self-coupling of a single nucleus [S] . We are trying to answer this question for Hz in section 3, using a simple ST0 approach. In section 4 we use CT0 calculations for studying the Z-dependence of the high frequency tail along the se.:es H-H, C-C, Si-Si. In this section we also want to reach a reasonable-agreement with experiment. Furthermore we compare the structures in the spectral density function. The SOS method wiil never exhaust the Hilbert space of the reEv,ant excited electronic states. The contribution from the omitted parts is expressed in terms of a “residual spectral density function” in section 5. We then use the method for analysing the completeness of &set of virtual orbitals of Kowalewski et al. 121. Such an analysis is particularly interesting because -their results’agrec with experiment within 0.5 HZ and can be expected to be converged. In order to obtain additional information about this density of states of the non-orthogonal basis set, we use it to calculate the polarizabilities of H and He in the appendix.
coupling
tion to the nuclear spin-spin coupling constant. Then the Ramsey theory gives for the reduced nuclear spinspin coupling constant K = IzJ@2
(1)
the expression
= 2.41457 X 1O-58 m6 kg2 ,A Ae2. The St dimension of K is rnM2 kg sm2 AB2_ 1 cms3 in the gauss-cgs system is 0.1 mV2 kg se2 A-’ in the SI system. Here the total ground state wavefunction is taken to be the Slater determinant \I’0 = $0)
~,(2)+5@01
and the excited state ‘k, = l@,(l) &(2).-$$2i
- 1) @Vi) ... &&LNIl.(4)
In a traditional sum-over-states approach 6’ would be one of the virtual orbitals 4, usually obtained while calculating the ground state wavefunction +o. In that case one has
If the sum, EC, has the value of 1 au, K = 2.5221 X 10lg me2 kg sb2 Ae2. For the triplet excitations involved. AE = ei - er f [iilrr).
(6)
Here Ei and er are the orbital energies of the two MOs and (iih) a Coulomb integral. In the present case we want to vary $’ continuously, writing ti: = P(1 L!N-“2
[x(l) + x(2)] *
(7)
Here the x are 1s orbitals of either gaussian or Slater type: x, = (2&r)3/4exp(-&),
2. Theory ’ .-. In this work we or& consider the contact oontribu-
(3)
xr = (t3 hF2exp(-+ and :
1
(8) 69
s =(x(1)1x(2)1
(20)
is the overlap integral between the two atomic orbitals at the centres 1 and 2. Then the spin-spin coupling constant becomes
B, = + exp(-2Q)/[
1 f expl-2Q)I
(21)
and K = C ‘g r cl{ D(c) (~ji16(r,)1~“)(~“16(r,)1~,)(~-1 i 0 (11) (for the gaussian osbitals replace { by ol). The state Q” is obtained from 9’ by orthogonalizing it against all the occupied orbitals Qi:
The required “density of states”, D(c), is found from the following heuristic argument: Consider a state I/cl) and another normalized state Ik) = (1 + b’)-“2(lk,)
+ blk2)),
where k,) is a state, orthogonal small number. The overlap (X’lk$ = (1 f b”)-‘;’
03) to !k,), and b is a
Z-Z1 - $$.
(14)
Suppose that k,) corresponds to an ST0 with orbital exponent < and Ik) to an ST0 with orbital exponent 5’ = 5 + At.
(1%
Then the overlap ({It’) = 8(tc’)‘/‘(5
+ c’)-3 = 1 - $(a
Thus the “amplitude the change A{, is
of new state” brought
(16)
= &r/d< =
for H,
In this chapter we investigate the high-frequency behaviour of the spectral density function a(c) for the hydrogen molecule, using a simple ST0 model. The ground slate wave function is assumed to be a single Slater determinant. see eq. (3). The molecular Q. is composed of one Slater type orbital, as defined in eq. (9), at each centre. The excited state is described by eq. (4), with the excited MO given in eq. (7). Writing explicitly the even and odd contributions, awe obtain from eq. (11) for the reduced coupling constant j
dJ-D(S‘)
(17)
Aliio.b/A[
+ G5016(rl )I~~IIx~~~s(r2)l~~o,/(~~ -
= 1.&2c.
For a single CT0 a similar argument fI(Cl) =iG
3. ST0 calculations
about by
and the number of states per unit < D(i)
where R is the internuclear distance. As the density of states is calculated for @‘, it is important to observe that the state 0” is not renormalized after the orthogonalization against the occupied orbital% eq. (I 2). A renormalization would lead to a spurious enhancement of the contribution of 0” when it is used in second-order perturbation theory involving continuous integration over Q.
K=C
b = &/%k%)
(22)
Q=&YR”,
(18) gives (19)
In section 3 we only shall discuss high values of c, for which eq. (18) is sufficient. In section 4, all possible values of (Ywill be needed and we have to repeat the argument above for the symmetric and antisymmetric excited MOs & and @_. This gives
E-II .
(233
E+ and E_ are the total energies for the even and odd triplet excited states, respectively. For high values of the orbital exprjnent, {‘, for the-scanning molecular orbital, both the two terms behave as (I’)“. However, the sum of the two terms gives for the “tail” contribution from the relativistic momentum region above cId:
2&C(l = --$-
+ e-R)4
i 1 +s
(Ll)-2*
(24)
276
..
H.B. hnsei
et al./Specrml
density anolvsis of ~~cicor spin-spin CoupIing
R is the internuclear distance and S is the overlap integral between the two ground state AOs, for which { = 1 is assumed. A ieasonable value for trel is 100 au. With R = 1.4 au we then have for Kd the value of 1 X 10v2. This is about 5 X lOA times the total coupling constant K and therefore the order of magnitude is about the same as for the relativistic correction to the ground state hyperfme matrix element for hydrogen.
4. CT0 calculations
very hi& a (up to 500 000). ?‘he&erna-tive possibility is that in those calculations the functions with high exponents are needed to get a more complete description of the set of excited states. We therefore have performed calculations on Hz using three different basis sets and the values for the reduced coupling constants calculated with these are given in table 1. The main point here is the stability of the results. With 4 gaussians at each hydrogen atom (set 1) we already get a value that is about 5% off from the converged value, that is obtained with basis sets 2 and 3. This converged value of K = 27.2 [lOI9 m-’ kg S* Aw2] still is around 16% too high. There are two reasons for a discrepancy between these values and the experimental ones. The first is that we have not taken into account any correlation. The second is that the density of states might deviate from our computed value. It is now also possible to find out how the contributlons vary as a function of 01.We therefore first look at the separate contributions from @+ and $_, plotted as a+ and a_. These are shown in fig. 1, both for the basis sets 1 and 3. For high (Ythe separate contributions behave as a*, but the sum as or-‘. The even scanning molecular orbital gives a minimum because of the or-
(25) which means that in @(IY)we also have included Now the integration over a can be performed ,ically.
(YD(a). numer-
4.1. Ttre hydrogen molecule. For-the hydrogen molecule there are many investigations with the SOS method and unless large basis sets are used,‘the resulJs are dinl~pointing [7]. It is often -assumed that this is caused by deficiencies in the ground state wave function, especially the density at the nuclei (81, cd then it is suggested that this is the reason why the. results are iinproved by including basisfunctions.wjth
lo-'
10-a lo-’
.,., ,-
.,
. .:
=
lo’
lo*
lo’
rol’
Fig. 1. Colttributions to the spectral density function @ for the hydrogen molecule from the even and odd “scanning molecular orbit&“, b+and o_, as a function of the orbital exponent Q. The dashed line corresponds to the results for the small basis set 1 and the full line to the results for the large basis set 3.
. -:
1
.I ., _
-.
MB. Jarfrefr er al.ISpecrral
derrsitv
anai_vsis of nuclear spin-spin
colrpii~lg
277
Table 1 Basis sets and results for Hz Basis se!
Uncontncted
1
(4) (8) @,3, 1)
2 3
Contracted
K (IO’9 m-* kg s-’ A-I)
(5, 3, DC’
25.7=) 27.24 27.16
‘) Experimcnlal value 23.3 X 1CJ’9m-’ kg s-’ A-‘. b, From ref. [9 1. ‘) Frum ref. [Z]. thogonalization to the ground state MO, but for lower values of (Yits contribution is larger. For low (Ythe even contribution behaves roughly as LY 3i2, the odd as 612. The difference can be understood by thinking of the even combination as an s-type function, and the odd combination as a p,-type orbital. Remarkable again is the small difference between the results obtained with the two basis sets. Apparently therefore the main cause of the differences in the results obtained with various SOS calculations must be the incompleteness of the set of excited states. 4.2. Acetylene and disiliron-di~l~ldride By using this same method we also can do calculations on other systems than Hz. As we are especially interested in the dependency of the high frequency tail on the nuclear charge Z, we now will discuss calculations performed on C,H, and Si,H,. The results of these calculations and the data about the basis sets used, are collected in table 2. We can compare the total computed value for C2H2 with experiment and then we find a difference of only about 13%. The reasons for the deviation must be the same Table 2 Basis sets at C and Si, obtained from ref. [9] and results for CaHp and .&HZ System
Basis
HZ
(9, 3, I) 4 (5, 3.1) (11.6) -+(4,2, (14,lO) -+ (4.3)
C2H2
Si2 Hi
Krel K .Kexp (lo-” ms2 kg s-2 Am2) 0.0059 27.2 23.3 0.066 195.9 225.9 0.101 251.0 -
as in the case of Hz. On the other hand these results are better than for many SOS calculations [lo] _In this same table also the results for a tail contribution are collected. As the high frequency tail contribution we will define, just as in section 3, the contribution from the region with Slater exponent { > 102, and therefore 01 > 104. In the ST0 calculation for the hydrogen molecule the contribution to K from the relativistic momentum region (the “tail’‘-contribution) was about of the same relative order of maqitude as the relativistic correction B(n. Z) (see ref. [ 111) for the ground state hyperfine matrix element. As the latter relativistic correction, B-l grows as Z’, it is interesting to ask how this tail contribution will behave as a function of Z. However, from the data in table 2 it is immediately clear that the situation here is different.-In each calculation the tail contribution is of the relative order 10e3 to lo4 and no Zdependence can be found in the case of this coupling. The absoIute value of the tail contribution just seems to depend on the value of K. Again we do not have to restrict our discussion to the numerical answer. We also can study the behaviour of the operand. At each value ofa we can form *(IX), the total sum of the contribution for this Q from all the combinations of an occupied orbital with the even or odd excited orbital corresponding to this 0~.For the three systems Hz (basis set 3), &Hz and SizH2 such plots of @(a) against LIare given in fig. 2. With the help of these pictures we first of all can decide if it is possible to use an average energy denominator in calculations for these cases with directly bonded atoms. In all three cases there is one main peak so from that point of view it should be possible to choose an average energy. On the other hand, the width of the peak is about one order of magnit:>de in Q, and this iesults in a change of
most of these cancel. The most important remaining contribution is that from the u-bonding orbital betw. :n the two atoms for which we calculate the coupling constant. We therefore apparently have to correlate the structure of this u-bonding orbital, which is orthogonal to core-orbitals. with the structures in the figures. This will be a subject of further study.
I
az-
-CL?-
Ax-
5. Residual -0.6 -
spectral
density
functions
HZ
10x*
,
I
clsy 0
-02 B -Q4-
-cu5-
1
Another possibility for applying the spectral density function approach is the analysis of the completeness of the set of virtuals in a sum over states calculation. When the scanning molecular orbital is orthogonalized not only to the occupied MOs but also to the set of virtual orbitals, the remaining or residual G will only give significant contributions for those parts of Hilbert space that are not emptied by the set of virtuals. One possible way of studying this is to look at the fig. 3 and 4 in which t& residual spectral density cI,,,(cr) is shown for a number of cases. When in these figures +,(a) deviates significantly from zero, this means that for that a-region the description with the virtual orbit& is incomplete.
02-
-02-
-cw-
-a-
S&l.
10“
lc-*
lo-'
1
10'
10'
roJ
(I
Kg. 2. Total spectral density functions @(a) for H2 (basis set 3). far the C-C coupling in CzH2. and for the Si-Si coupling in Si2 Hz.
one order of magnitude for the corresponding orbital energies too. Furthermore there definitely exists a structure, with pckitive contributions in one region and negative in anOt+.
&alysing &is structure is another question. By inspection of the separate, contributions it appears that
Fig. 3. The residual spectral density functions lor the hydrogen molecule. The full line depicts ares(~) for the basis set 1 and a sin& virtual orbital. The d&ted Line depicts @,,,(a) for the small basis set number 1, including all virtual orbit&. For comparison, we show the full spectrd densily function @(a) for the basis set I ti a dashed lipe:
.’ ..
Fig. 4. The residual spectral density for the hydrogen molecule with the large basis set 3. The full line depicts @,,&a). For comparison O(a) for this basis set is given as a dashed line.
From fig. 3 we see that a description with only one virtual, obtained as virtual in a calculation with the 4 basis functions of basis set 1 contracted to 1, is completely insufficient. The ares(~) is comparable to @(a) itself, but shifted. With 4 virtuals the description is clearly improved. This can also be concluded from the value Of R,, the integrated value of @&a), which is now: = 0.6, compared with 22.0 for the contracted calG culation. A similar situation occurs with basis set 3. Here again the description with the set of virtual orbitals is better than in the case of the smaller basis set. Now there are only two regions where Q -(a) deviates significantly from zero: around (Y= IO-‘9 and around (Y= 3. The total value of the integrated @,,(a) is small: K = 1.0. This value is so small because of a cancellatiri of the contributions from the positive and the negative region. When we extend and improve the set of virtuals by no longer contracting the Is CT0 functions in the basis set, the value of Q&or) around Q = I 0e2 becomes
much smaller,
proved,
the description
but the value of K,, becomes no cancellation occurs (K, = 2.3).
there is im-
larger because
6. Conclusions In part I of this series it is found that for the cou-
pling constant JxH in a series of molecules XH4 the experimental trends can be reproduced using a HulthCn potential model. In that case the calculation by way of integration over all excited states is straightforward. For the systems we consider here, with homonuclear coupling, this integration also appears to be a well functioning alternative for the sum over states approach. With the density of states as defined here from differential arguments, and further tested in the appendix, this approach leads to good results. The +40% deviation for the hydrogen atom, and the -30% for helium indicate the reliability of the density of states as defined here, although it must be realized that only the hydrogen atom ST0 calculation is exact. For the other cases the description of the ground state - and for helium also the excited state - wave function is only approximate. With this scanning molecular orbital method results for Hz and for CzH2 are found that show a similar correspondence with the experimental values. An important conCiusion follows from the stability of the results for the hydrogen molecule with various basis sets. From that the conclusion can be drawn that in the calculation of the coupling constant with the SOS method the sometimes very bad results are due to inadequacies in the set of virtuals and not in the first place to wrong values for the electron density at the nuclei. Further evidence for this conclusion comes from the analysis with the residual spectral density function that even indicates what a-values, and therefore what parts of the Hilbert space of the excited triplets, are not sufficiently covered with the virtual orbitals. Of course such an incompleteness can for a large basis set calculation rather easily be removed when both ground state and excited states are formed in configuration interaction calculation. Finally the conclusions for the high energy tail are the same as in part I. The high energy part or high momentum part of the spectral density corresponding with 5 > IO2 is of the order of low3 to lOA of the total coupling constant and no functional relation can be found with the nuclear charge. This is in contrast to the nuclear self.coupling constant of a hydrogen-like atom where the ultra relativistic continuum is very important, and to the relativistic correction B(n. Z) which behaves like Z2 for small 2.
280
Acknkvledgement
a’.?”= i 6&F2b3 j= dt’[({‘)4/(c Part of this iork was carried out during the visits of H-B. Jansen at the University ofJyviskyl% and of P. Pyykkb at the Free University, Amsterdam. We wish to thank the& institution for supporting the callaboration. H.B. Jan&n acknowledges the support of his stay at C.E.C.A.M., Orsay, France by the Netherlands Foundation for Chemical Research (SON) with financial aid from the Netherlands Organization for the Advancement of.Pure Research (ZWO).
Appendix In this appendix we calculate the electric pblarizabilities of the H and He atoms using the same scanning orbital approach. !n this way we can obtain additional information about the appiicability of the method. For the Is state of a hydrogen-like atom with nuclear charge 2 the perturbation H’=Fz,
(Al)
where F is the electric. field, gives a quadratic contribution [ 121 ti’)
= -(Y/4_+)
energy
F’_
642)
The excited state is taken to be a p-state of either gaussian or Slater type: $(r) c+
5 2&(
,
2cr/1r)3/4 eG2
.x;(r) =[({‘)5”/2+G] co@ The relevant density-of-states DP(&) = b-fia’, DP(S) = i$
re+‘*.
(A4)
functions
then become
.
(As)
c-1.
(Ah)
The second+rder @)
iw
energies
= 2fiOF2a312o
r
then become
da’[(&)3’2/(a0
+ or’)‘]
0
X I$,+, -I-2zw
- 3~‘. +
~ZJ%%TI-~,
and
: .:
:
.. :
: :
CA71
+ f’/2)‘“]
0
x [-fz2 + &Z{’ - +(fy] -I
7
W)
where a0 is the ground state Is CT0 exponent, set equal to 0.271 c2_ Putting Z = t = 1 we get for the hydrogen atom the polarizabilities Q: CTO:
cu=.+‘/~‘=3.13
(LQ,
(A93
STO:
(Y= ‘@)/ti
= 3.22 (a,)3,
IA101
where no is the Bohr radius. These results are too high by 39 and 43 per cent, respectively. For the helium atom, using one ST0 with < = 1.6875 for the ground state description we similarly get Q = d”/F’
= 0.98
(u~)~
(All)
which result is now 29% too low.
References [ 11 W. Meyer, Z. Physik 229 (1969) 452. [2] J. Mowalewski, B. Roos. P. Siegbahn and R. Vestin. Chem. Phys. 3 (1974) 70. [3] P. Pyykkij and J. Jokiwari, Chem. Phys. 10 (1975) 293. [4] P.M. Getzin, Ph.D. Thesis, Columbia University (1967); Diss. Abstr. 628 (1968) 4087. [S] E. Latvamaa, L. Kurittu. P. Pyykkij and L. Tntnru. J. Phys. B. 6 (1973) 591. [6] T. Kato, Commun. Pure Appl. Math. 10 (1957) 151. [7] P. Ros, in: Selected Topics in hlolecular Physics, ed. E. Clementi (Verlag Chemie, 1972). [S] J.M. Sch,ulman and D.N. Kaufman, I. Chem. Phys. 53 (2970) 477. [9] ~0 S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. b) S. Huzinaga and Y. Sakai, I. Chem. Phys. 50 (1969) 1371. c) S. Huzinaga and C. Arnau, J. Chem. Phys. 52 (1970) 2224. [IO] E-A-G. Armour and A.J. Stone, Proc. Roy. Sot. A302 (1967) 25. [ 1 l] P. Pyykkii, E. Pajanne and hi. Inokuti, Internat. J. Quantum Chem. 7 (1473) 785. 1121 H-A. Bethe and E.E. Salpeter; Quantum mechanics of one- and two-electron atoms (Springer Verlag. Berlin, 1957) p_ 232 ff.
SPECTRAL
DENSITY
II. Bartree-Fock
H.B. JANSEN,
LCAO
ANALYSIS
OF NUCLEAR
studies for homonuclear
SPIN-SPIN
COUPLlNG
coupling constants*
J.A.B. LOHMAN
Received 26 hky 1975
A spectral density function has been calculated for the indirect nuclear spin-spin coupling constant Tar homonuclar coupling. The ground state wavefunction is obtained with a normal ab-initio nlculation. The sum over states approach for cnlculating the reduced coupling constant K is replaced by an integation over a spectral density function-where the integration variable is the orbital exponent of a “scanning molecular orbits’. This results in n stable method for calculating K with reasonable accuracy. The spectral density function also gives information about which excited &es give important contributions to K. lzurthermore a residual spectral density function is defined that cm be used as a test for the completeness ot’a set ofvirtunl orbit& in n sum over states calculation.
1. introduction
Accurate numerical calculations of nuclear spinspin coupling constants in molecules have proved difficult. Perhaps the first and only quantitative calculations of a spin-spin coupling constant, with an estimated error of 10% or less, are those of Meyer [I] and Kowalewski et al. [2] for HZ. Ground state wavefunctions, approaching the Hartree-Fock level, can presently be obtained for medium-sized molecules. However, the spin-spin coupling is a second-order property, involving an infinite summation over the Hilbert space of all excited electronic states of the molecule. In practice, rather few excited states may suffice if they are judiciously chosen, E.g., for H2, the single virtual MO * This work was begun when H.B.J. and P.P. were at CECAM. Bstiment 506. Universitdde Ptis XI. Orsay, France. ** Present address: Department of Physical Chemistry, Abo Akademi, Finland.
of a simple LC.40 model can give a satisfactory agreement with experiment. The main thing, then, is knowing what are these important parts of the Hilbert space. One approach to this problem is presented in ref. [3] (part I of this series) for the X-H coupling constants in hydrides XHq. There the excited state? are rcpresented in terms of an orthonormal set of continuum wavefunctions in a Huhhen screened Coulomb potential describing the heavy atom, X. In the present work we consider homonuclear coupling constants. Not having an equally neat, orthonormal and continuous basis set in the present case, we describe the excited states of the molecule with a symmetric or antisymmetric molecular orbital, consisting,of two atomic orbitals, of gaussian or Slater type, treating the orbital exponent as a continuous variable. This basis is complete, in a subspace of the Hilbert space, but it is nonorthogonal.. The relevant densities of states are discussed in section 2. The total spin-spin coupling constant is then expressed as a continuous integral over the orbital expo-
274
H.B. fanSen ef aI.lSpecml
density analysis of nuclear spin-spin
nent. The integrand provides a spectral density analysis of the coupling constant. The integration over the orbital exponent is essentially equivalent to an integration in momentum space. Our method could thus be viewed as a continuous version of the sum over states (SQS) approach. The idea of varying continuously one additional A0 orbital exponent has been used by Getzin [4], but only in order to investigate the stability of the discrete SOS. The present method is not a very accurate one, but the purpose of the work is to get a feeling for thi: shape of the spectral density function. Therefore correlation effects are not considered in this work. One purpose of this work, then, is to get a feeling for the important parts of the Hilbert space of excited electronic states. This could be helpful in assigning the deficiency of any particular virtual orbital calculation to errors in the ground state wave function or to incompleteness of the set of excited states. Secondly, while the nuclear spin-spin coupling mainly is a lowfrequency polarizabibty, one may ask whether a highfrequency tail would exist like For the self-coupling of a single nucleus [S] . We are trying to answer this question for Hz in section 3, using a simple ST0 approach. In section 4 we use CT0 calculations for studying the Z-dependence of the high frequency tail along the se.:es H-H, C-C, Si-Si. In this section we also want to reach a reasonable-agreement with experiment. Furthermore we compare the structures in the spectral density function. The SOS method wiil never exhaust the Hilbert space of the reEv,ant excited electronic states. The contribution from the omitted parts is expressed in terms of a “residual spectral density function” in section 5. We then use the method for analysing the completeness of &set of virtual orbitals of Kowalewski et al. 121. Such an analysis is particularly interesting because -their results’agrec with experiment within 0.5 HZ and can be expected to be converged. In order to obtain additional information about this density of states of the non-orthogonal basis set, we use it to calculate the polarizabilities of H and He in the appendix.
coupling
tion to the nuclear spin-spin coupling constant. Then the Ramsey theory gives for the reduced nuclear spinspin coupling constant K = IzJ@2
(1)
the expression
= 2.41457 X 1O-58 m6 kg2 ,A Ae2. The St dimension of K is rnM2 kg sm2 AB2_ 1 cms3 in the gauss-cgs system is 0.1 mV2 kg se2 A-’ in the SI system. Here the total ground state wavefunction is taken to be the Slater determinant \I’0 = $0)
~,(2)+5@01
and the excited state ‘k, = l@,(l) &(2).-$$2i
- 1) @Vi) ... &&LNIl.(4)
In a traditional sum-over-states approach 6’ would be one of the virtual orbitals 4, usually obtained while calculating the ground state wavefunction +o. In that case one has
If the sum, EC, has the value of 1 au, K = 2.5221 X 10lg me2 kg sb2 Ae2. For the triplet excitations involved. AE = ei - er f [iilrr).
(6)
Here Ei and er are the orbital energies of the two MOs and (iih) a Coulomb integral. In the present case we want to vary $’ continuously, writing ti: = P(1 L!N-“2
[x(l) + x(2)] *
(7)
Here the x are 1s orbitals of either gaussian or Slater type: x, = (2&r)3/4exp(-&),
2. Theory ’ .-. In this work we or& consider the contact oontribu-
(3)
xr = (t3 hF2exp(-+ and :
1
(8) 69
s =(x(1)1x(2)1
(20)
is the overlap integral between the two atomic orbitals at the centres 1 and 2. Then the spin-spin coupling constant becomes
B, = + exp(-2Q)/[
1 f expl-2Q)I
(21)
and K = C ‘g r cl{ D(c) (~ji16(r,)1~“)(~“16(r,)1~,)(~-1 i 0 (11) (for the gaussian osbitals replace { by ol). The state Q” is obtained from 9’ by orthogonalizing it against all the occupied orbitals Qi:
The required “density of states”, D(c), is found from the following heuristic argument: Consider a state I/cl) and another normalized state Ik) = (1 + b’)-“2(lk,)
+ blk2)),
where k,) is a state, orthogonal small number. The overlap (X’lk$ = (1 f b”)-‘;’
03) to !k,), and b is a
Z-Z1 - $$.
(14)
Suppose that k,) corresponds to an ST0 with orbital exponent < and Ik) to an ST0 with orbital exponent 5’ = 5 + At.
(1%
Then the overlap ({It’) = 8(tc’)‘/‘(5
+ c’)-3 = 1 - $(a
Thus the “amplitude the change A{, is
of new state” brought
(16)
= &r/d< =
for H,
In this chapter we investigate the high-frequency behaviour of the spectral density function a(c) for the hydrogen molecule, using a simple ST0 model. The ground slate wave function is assumed to be a single Slater determinant. see eq. (3). The molecular Q. is composed of one Slater type orbital, as defined in eq. (9), at each centre. The excited state is described by eq. (4), with the excited MO given in eq. (7). Writing explicitly the even and odd contributions, awe obtain from eq. (11) for the reduced coupling constant j
dJ-D(S‘)
(17)
Aliio.b/A[
+ G5016(rl )I~~IIx~~~s(r2)l~~o,/(~~ -
= 1.&2c.
For a single CT0 a similar argument fI(Cl) =iG
3. ST0 calculations
about by
and the number of states per unit < D(i)
where R is the internuclear distance. As the density of states is calculated for @‘, it is important to observe that the state 0” is not renormalized after the orthogonalization against the occupied orbital% eq. (I 2). A renormalization would lead to a spurious enhancement of the contribution of 0” when it is used in second-order perturbation theory involving continuous integration over Q.
K=C
b = &/%k%)
(22)
Q=&YR”,
(18) gives (19)
In section 3 we only shall discuss high values of c, for which eq. (18) is sufficient. In section 4, all possible values of (Ywill be needed and we have to repeat the argument above for the symmetric and antisymmetric excited MOs & and @_. This gives
E-II .
(233
E+ and E_ are the total energies for the even and odd triplet excited states, respectively. For high values of the orbital exprjnent, {‘, for the-scanning molecular orbital, both the two terms behave as (I’)“. However, the sum of the two terms gives for the “tail” contribution from the relativistic momentum region above cId:
2&C(l = --$-
+ e-R)4
i 1 +s
(Ll)-2*
(24)
276
..
H.B. hnsei
et al./Specrml
density anolvsis of ~~cicor spin-spin CoupIing
R is the internuclear distance and S is the overlap integral between the two ground state AOs, for which { = 1 is assumed. A ieasonable value for trel is 100 au. With R = 1.4 au we then have for Kd the value of 1 X 10v2. This is about 5 X lOA times the total coupling constant K and therefore the order of magnitude is about the same as for the relativistic correction to the ground state hyperfme matrix element for hydrogen.
4. CT0 calculations
very hi& a (up to 500 000). ?‘he&erna-tive possibility is that in those calculations the functions with high exponents are needed to get a more complete description of the set of excited states. We therefore have performed calculations on Hz using three different basis sets and the values for the reduced coupling constants calculated with these are given in table 1. The main point here is the stability of the results. With 4 gaussians at each hydrogen atom (set 1) we already get a value that is about 5% off from the converged value, that is obtained with basis sets 2 and 3. This converged value of K = 27.2 [lOI9 m-’ kg S* Aw2] still is around 16% too high. There are two reasons for a discrepancy between these values and the experimental ones. The first is that we have not taken into account any correlation. The second is that the density of states might deviate from our computed value. It is now also possible to find out how the contributlons vary as a function of 01.We therefore first look at the separate contributions from @+ and $_, plotted as a+ and a_. These are shown in fig. 1, both for the basis sets 1 and 3. For high (Ythe separate contributions behave as a*, but the sum as or-‘. The even scanning molecular orbital gives a minimum because of the or-
(25) which means that in @(IY)we also have included Now the integration over a can be performed ,ically.
(YD(a). numer-
4.1. Ttre hydrogen molecule. For-the hydrogen molecule there are many investigations with the SOS method and unless large basis sets are used,‘the resulJs are dinl~pointing [7]. It is often -assumed that this is caused by deficiencies in the ground state wave function, especially the density at the nuclei (81, cd then it is suggested that this is the reason why the. results are iinproved by including basisfunctions.wjth
lo-'
10-a lo-’
.,., ,-
.,
. .:
=
lo’
lo*
lo’
rol’
Fig. 1. Colttributions to the spectral density function @ for the hydrogen molecule from the even and odd “scanning molecular orbit&“, b+and o_, as a function of the orbital exponent Q. The dashed line corresponds to the results for the small basis set 1 and the full line to the results for the large basis set 3.
. -:
1
.I ., _
-.
MB. Jarfrefr er al.ISpecrral
derrsitv
anai_vsis of nuclear spin-spin
colrpii~lg
277
Table 1 Basis sets and results for Hz Basis se!
Uncontncted
1
(4) (8) @,3, 1)
2 3
Contracted
K (IO’9 m-* kg s-’ A-I)
(5, 3, DC’
25.7=) 27.24 27.16
‘) Experimcnlal value 23.3 X 1CJ’9m-’ kg s-’ A-‘. b, From ref. [9 1. ‘) Frum ref. [Z]. thogonalization to the ground state MO, but for lower values of (Yits contribution is larger. For low (Ythe even contribution behaves roughly as LY 3i2, the odd as 612. The difference can be understood by thinking of the even combination as an s-type function, and the odd combination as a p,-type orbital. Remarkable again is the small difference between the results obtained with the two basis sets. Apparently therefore the main cause of the differences in the results obtained with various SOS calculations must be the incompleteness of the set of excited states. 4.2. Acetylene and disiliron-di~l~ldride By using this same method we also can do calculations on other systems than Hz. As we are especially interested in the dependency of the high frequency tail on the nuclear charge Z, we now will discuss calculations performed on C,H, and Si,H,. The results of these calculations and the data about the basis sets used, are collected in table 2. We can compare the total computed value for C2H2 with experiment and then we find a difference of only about 13%. The reasons for the deviation must be the same Table 2 Basis sets at C and Si, obtained from ref. [9] and results for CaHp and .&HZ System
Basis
HZ
(9, 3, I) 4 (5, 3.1) (11.6) -+(4,2, (14,lO) -+ (4.3)
C2H2
Si2 Hi
Krel K .Kexp (lo-” ms2 kg s-2 Am2) 0.0059 27.2 23.3 0.066 195.9 225.9 0.101 251.0 -
as in the case of Hz. On the other hand these results are better than for many SOS calculations [lo] _In this same table also the results for a tail contribution are collected. As the high frequency tail contribution we will define, just as in section 3, the contribution from the region with Slater exponent { > 102, and therefore 01 > 104. In the ST0 calculation for the hydrogen molecule the contribution to K from the relativistic momentum region (the “tail’‘-contribution) was about of the same relative order of maqitude as the relativistic correction B(n. Z) (see ref. [ 111) for the ground state hyperfine matrix element. As the latter relativistic correction, B-l grows as Z’, it is interesting to ask how this tail contribution will behave as a function of Z. However, from the data in table 2 it is immediately clear that the situation here is different.-In each calculation the tail contribution is of the relative order 10e3 to lo4 and no Zdependence can be found in the case of this coupling. The absoIute value of the tail contribution just seems to depend on the value of K. Again we do not have to restrict our discussion to the numerical answer. We also can study the behaviour of the operand. At each value ofa we can form *(IX), the total sum of the contribution for this Q from all the combinations of an occupied orbital with the even or odd excited orbital corresponding to this 0~.For the three systems Hz (basis set 3), &Hz and SizH2 such plots of @(a) against LIare given in fig. 2. With the help of these pictures we first of all can decide if it is possible to use an average energy denominator in calculations for these cases with directly bonded atoms. In all three cases there is one main peak so from that point of view it should be possible to choose an average energy. On the other hand, the width of the peak is about one order of magnit:>de in Q, and this iesults in a change of
most of these cancel. The most important remaining contribution is that from the u-bonding orbital betw. :n the two atoms for which we calculate the coupling constant. We therefore apparently have to correlate the structure of this u-bonding orbital, which is orthogonal to core-orbitals. with the structures in the figures. This will be a subject of further study.
I
az-
-CL?-
Ax-
5. Residual -0.6 -
spectral
density
functions
HZ
10x*
,
I
clsy 0
-02 B -Q4-
-cu5-
1
Another possibility for applying the spectral density function approach is the analysis of the completeness of the set of virtuals in a sum over states calculation. When the scanning molecular orbital is orthogonalized not only to the occupied MOs but also to the set of virtual orbitals, the remaining or residual G will only give significant contributions for those parts of Hilbert space that are not emptied by the set of virtuals. One possible way of studying this is to look at the fig. 3 and 4 in which t& residual spectral density cI,,,(cr) is shown for a number of cases. When in these figures +,(a) deviates significantly from zero, this means that for that a-region the description with the virtual orbit& is incomplete.
02-
-02-
-cw-
-a-
S&l.
10“
lc-*
lo-'
1
10'
10'
roJ
(I
Kg. 2. Total spectral density functions @(a) for H2 (basis set 3). far the C-C coupling in CzH2. and for the Si-Si coupling in Si2 Hz.
one order of magnitude for the corresponding orbital energies too. Furthermore there definitely exists a structure, with pckitive contributions in one region and negative in anOt+.
&alysing &is structure is another question. By inspection of the separate, contributions it appears that
Fig. 3. The residual spectral density functions lor the hydrogen molecule. The full line depicts ares(~) for the basis set 1 and a sin& virtual orbital. The d&ted Line depicts @,,,(a) for the small basis set number 1, including all virtual orbit&. For comparison, we show the full spectrd densily function @(a) for the basis set I ti a dashed lipe:
.’ ..
Fig. 4. The residual spectral density for the hydrogen molecule with the large basis set 3. The full line depicts @,,&a). For comparison O(a) for this basis set is given as a dashed line.
From fig. 3 we see that a description with only one virtual, obtained as virtual in a calculation with the 4 basis functions of basis set 1 contracted to 1, is completely insufficient. The ares(~) is comparable to @(a) itself, but shifted. With 4 virtuals the description is clearly improved. This can also be concluded from the value Of R,, the integrated value of @&a), which is now: = 0.6, compared with 22.0 for the contracted calG culation. A similar situation occurs with basis set 3. Here again the description with the set of virtual orbitals is better than in the case of the smaller basis set. Now there are only two regions where Q -(a) deviates significantly from zero: around (Y= IO-‘9 and around (Y= 3. The total value of the integrated @,,(a) is small: K = 1.0. This value is so small because of a cancellatiri of the contributions from the positive and the negative region. When we extend and improve the set of virtuals by no longer contracting the Is CT0 functions in the basis set, the value of Q&or) around Q = I 0e2 becomes
much smaller,
proved,
the description
but the value of K,, becomes no cancellation occurs (K, = 2.3).
there is im-
larger because
6. Conclusions In part I of this series it is found that for the cou-
pling constant JxH in a series of molecules XH4 the experimental trends can be reproduced using a HulthCn potential model. In that case the calculation by way of integration over all excited states is straightforward. For the systems we consider here, with homonuclear coupling, this integration also appears to be a well functioning alternative for the sum over states approach. With the density of states as defined here from differential arguments, and further tested in the appendix, this approach leads to good results. The +40% deviation for the hydrogen atom, and the -30% for helium indicate the reliability of the density of states as defined here, although it must be realized that only the hydrogen atom ST0 calculation is exact. For the other cases the description of the ground state - and for helium also the excited state - wave function is only approximate. With this scanning molecular orbital method results for Hz and for CzH2 are found that show a similar correspondence with the experimental values. An important conCiusion follows from the stability of the results for the hydrogen molecule with various basis sets. From that the conclusion can be drawn that in the calculation of the coupling constant with the SOS method the sometimes very bad results are due to inadequacies in the set of virtuals and not in the first place to wrong values for the electron density at the nuclei. Further evidence for this conclusion comes from the analysis with the residual spectral density function that even indicates what a-values, and therefore what parts of the Hilbert space of the excited triplets, are not sufficiently covered with the virtual orbitals. Of course such an incompleteness can for a large basis set calculation rather easily be removed when both ground state and excited states are formed in configuration interaction calculation. Finally the conclusions for the high energy tail are the same as in part I. The high energy part or high momentum part of the spectral density corresponding with 5 > IO2 is of the order of low3 to lOA of the total coupling constant and no functional relation can be found with the nuclear charge. This is in contrast to the nuclear self.coupling constant of a hydrogen-like atom where the ultra relativistic continuum is very important, and to the relativistic correction B(n. Z) which behaves like Z2 for small 2.
280
Acknkvledgement
a’.?”= i 6&F2b3 j= dt’[({‘)4/(c Part of this iork was carried out during the visits of H-B. Jansen at the University ofJyviskyl% and of P. Pyykkb at the Free University, Amsterdam. We wish to thank the& institution for supporting the callaboration. H.B. Jan&n acknowledges the support of his stay at C.E.C.A.M., Orsay, France by the Netherlands Foundation for Chemical Research (SON) with financial aid from the Netherlands Organization for the Advancement of.Pure Research (ZWO).
Appendix In this appendix we calculate the electric pblarizabilities of the H and He atoms using the same scanning orbital approach. !n this way we can obtain additional information about the appiicability of the method. For the Is state of a hydrogen-like atom with nuclear charge 2 the perturbation H’=Fz,
(Al)
where F is the electric. field, gives a quadratic contribution [ 121 ti’)
= -(Y/4_+)
energy
F’_
642)
The excited state is taken to be a p-state of either gaussian or Slater type: $(r) c+
5 2&(
,
2cr/1r)3/4 eG2
.x;(r) =[({‘)5”/2+G] co@ The relevant density-of-states DP(&) = b-fia’, DP(S) = i$
re+‘*.
(A4)
functions
then become
.
(As)
c-1.
(Ah)
The second+rder @)
iw
energies
= 2fiOF2a312o
r
then become
da’[(&)3’2/(a0
+ or’)‘]
0
X I$,+, -I-2zw
- 3~‘. +
~ZJ%%TI-~,
and
: .:
:
.. :
: :
CA71
+ f’/2)‘“]
0
x [-fz2 + &Z{’ - +(fy] -I
7
W)
where a0 is the ground state Is CT0 exponent, set equal to 0.271 c2_ Putting Z = t = 1 we get for the hydrogen atom the polarizabilities Q: CTO:
cu=.+‘/~‘=3.13
(LQ,
(A93
STO:
(Y= ‘@)/ti
= 3.22 (a,)3,
IA101
where no is the Bohr radius. These results are too high by 39 and 43 per cent, respectively. For the helium atom, using one ST0 with < = 1.6875 for the ground state description we similarly get Q = d”/F’
= 0.98
(u~)~
(All)
which result is now 29% too low.
References [ 11 W. Meyer, Z. Physik 229 (1969) 452. [2] J. Mowalewski, B. Roos. P. Siegbahn and R. Vestin. Chem. Phys. 3 (1974) 70. [3] P. Pyykkij and J. Jokiwari, Chem. Phys. 10 (1975) 293. [4] P.M. Getzin, Ph.D. Thesis, Columbia University (1967); Diss. Abstr. 628 (1968) 4087. [S] E. Latvamaa, L. Kurittu. P. Pyykkij and L. Tntnru. J. Phys. B. 6 (1973) 591. [6] T. Kato, Commun. Pure Appl. Math. 10 (1957) 151. [7] P. Ros, in: Selected Topics in hlolecular Physics, ed. E. Clementi (Verlag Chemie, 1972). [S] J.M. Sch,ulman and D.N. Kaufman, I. Chem. Phys. 53 (2970) 477. [9] ~0 S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. b) S. Huzinaga and Y. Sakai, I. Chem. Phys. 50 (1969) 1371. c) S. Huzinaga and C. Arnau, J. Chem. Phys. 52 (1970) 2224. [IO] E-A-G. Armour and A.J. Stone, Proc. Roy. Sot. A302 (1967) 25. [ 1 l] P. Pyykkii, E. Pajanne and hi. Inokuti, Internat. J. Quantum Chem. 7 (1473) 785. 1121 H-A. Bethe and E.E. Salpeter; Quantum mechanics of one- and two-electron atoms (Springer Verlag. Berlin, 1957) p_ 232 ff.