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Electronic structure of diborane and hyperconjugation effect
JOVRN.\L
OF MOLECULAR
SPECTROSCOPY
21, 29-41
(1966)
Electronic Structure of Di borane and Hyperconjugation Effect TEIICHIRO Department
OGAWA AND Kozo
sf Chemistry, Faculty of Scieme,
Osaka
HIROT_~
lJniz)ersity,
Toyonaka,
Osaka, Japan
The ultraviolet
spectra of dihorane are re-examined and its electronic strucby the method of LCAO SCF. The hyperconjugat,ion effect of the bridge hydrogens is taken into consideration and t,he system is treated ns :t three-center, two-electron problem. The int,egrals are mainly calculated nonempirically but some of them are evaluated by making some approximation. The transit.ion energies are evaluated to be 6.7 (forbidden) and 10.5 e\ (allowed), and they correspond web to the experimental values of 6.9 (weak) and 9.2 eV (strong), respectively. The hyperconjngation effect is proved to play an important role in siirh a bridge compound. The nature of the bridge bond is discussed.
1ure is calculated
I. INTRODUCTIOS
The peculiar structures of boron hydrides have invited many investigators in bhis field. Diborane, the simplest of these hydrides, was concluded to he in a bridged st’ructure and two hydrogens of diborane lie between two boron atoms. TTarious interpretations were proposed to elucidate the structure of this molecule (I). Ogawa and Miyazawa (2) have recently calculated the normal vihrations of diborane and showed several characteristics of the bond in the bridge. Some type of bond was concluded to exist between two boron atoms through the analysis of the potential function, but’ no detailed explanation of bond character n-as proposed. Mulliken (S) discussed the MO electronic configuration for both ethane-like and bridge structure, and Yamazaki (4) calculated the self-consistent field molecular orbitals in LCAO forms for the ground state of diborane by Roothaan’s procedure. Hoffmann and Lipscomb (5) investigated the whole series of boron hydrides in t’erms of the LCAO treatment. The electronic absorption spectra of diborartc were measured by Rlum et al. (6) and Price (7). Three bands were found at lS20 i (6.S eT’ weak), 13.50 8 (9.3 eV strong), and 1200 A (10.3 eT_. strong). Little attempts, however, have been made to interpret these bands in terms of the LCAO treatment. Nethods of calculation have been developed on the electronic structure of various molecules quite satisfnctorilp. Many complex molecules, especially 29
30
OGAWA
AND
HIROTA
unsaturated hydrocarbons, have been made the objects of study, and their electronic spectra have been elucidated in terms of molecular energy levels. Hanazaki, Hosoya, and Nagakura (8) studied the electronic structure of the tbutyl cation and showed the importance of hyperconjugation effect in the molecules without any unsaturated bonds. Diborane also has no typical double bond, but two bridge hydrogens are situated as suitable for hyperconjugation with two vacant 2p7r orbitals of borons. In the present study, the spectrum was re-examined and the electronic structure of diborane is discussed in terms of the self-consistent field molecular orbitals in LCAO forms and the hyperconjugation effects of bridge hydrogens with two borons. The object of the present study is to assign the observed absorption bands to corresponding transitions of molecular energy levels and to clarify the nature of the bridge bond of this molecule. II. EXPERIMENTAL
Diborane was synthesized by adding sodium borohydride slowly to cold concent,rated sulfuric acid (9). In order to separate diborane from hydrogen, hydrogen sulfide, and sulfur dioxide which were reported to be produced as byproduct’s, the products were cooled down with liquid nitrogen and were evacuated. Then the liquid nitrogen was replaced by dry ice-methanol and the evolved gas was collected in a Toepler pump. Hydrogen sulfide was found in the sample to be 0.8 per cent and a trace of sulfur dioxide was found by the mass-spectrometric analysis. The ultraviolet spectra were measured by a Hitachi EPS-2 spectrophotometer in the 3400-2200 A region, by a Cary 14 PM spectrophotometer in the 22001900 8 region, and by a vacuum ultraviolet spectrometer at the Institute for Solid State Physics in the 2050-1550 i region. III.
EXPERIMENTAL
RESULTS
There was no absorption band in the 3400-1900 _®ion. The observed spectrum in the vacuum ultraviolet region is shown in Fig. 1. An absorption band was located at 1790 1 and its extinction coefficient was about 90. The increase of absorbance was found from 1650 i to the shorter wavelength region, but no absorption peak would be observed. The absorption band at 1790 A is assigned to a band forbidden ,by a symmetry selection rule because of its weak intensity. The band at 1350 A found by Price is perhaps the first allowed transition, and this band is tentatively assigned to the T -+ n* transition. IV. GENERAL
OUTLINE
OF THE
CALCULATION
The structure and the molecular dimensions of diborane were determined by the electron diffraction study (10) and are shown in Fig. 2. TWO bridge hydrogens lie in a plane normal to the plane of four terminal hydrogens. The bonds to the four terminal hydrogens are considered essentially single bonds and are as-
ELECTRONIC
FIG. 1. The
electronic
FIG. 2. The structure
spectrllm
of diborane.
STRUCTURE
of diborane
31
OF DIBORANE
in the vac,lum
ultraviolet
region
n = 1.19G d, 6 = 1.339 d, I, = 1.775 A, and a =
119.0”.
sumed to be sufficiently localized so that t’he bridge bonds are separated from t’he remaining part of the molecule (11). The terminal ( HBH angle is 122” and the B-B distance (1.775 A) is approximately equal to t,he sum (1.76 8) of Pauling’s tetrahedral covalent radii (1.2). Accordingly each boron atom is regarded to have the sp2 hybrid orbitals to the other boron atom as well as to t’wo t’erminal hydrogens and to have a vacant 2pprr
32
OGAWA
AND
HIROTA
orbital normal to the hybrid orbitals. Two bridge hydrogen atoms are treated as a united atom which has a pseudo u orbital x0 and a pseudo r orbital x1 :
x0 = M2U + SO>l’/2(UI + 4, Xl =
l/Lw
-
So>l’/2(al
-
(1)
f-321,
where al and aa are 1s orbitals of the bridge hydrogens. Similar assumptions are often adopted in the treatment of hyperconjugation of the methyl group (8, IS). Two 2pa orbitals of the borons (xZ and x3) and the x1 orbital of t’he bridge hydrogens interact with each other and make a T molecular orbital like that of ally1 radical (see Fig. 3). Resemblance in molecular structure of t’his compound to ethylene may allow the present treatment. Two .s@’ orbitals of borons make a B-B bond, with which the ~0 orbital of the bridge hydrogens will interact, and they make bonding and antibonding u orbitals. There are four electrons in these bridge bonds; two come from borons and two from bridge hydrogens. These four electrons are situated in the lowest and the second lowest orbitals. The lowest orbital must be the lowest bonding u orbit,al, B-HZ-B. The second lowest orbital is concluded not to be the second 0 orbital but to be a bonding a orbital, since the result of Hamilton (11) and the result of the nonempirical calculation by Yamazaki (4) show that the a-like orbital 47 (in Yamazaki’s notation) is very stable energetically. That is to say, in the bridge of diborane two electrons lie in the v orbital and the remaining two electrons in the P orbital; the situation is just like that of ethylene. In order to interpret the ultraviolet spectra, the energies of these 7relectrons are to be calculated. Only the energies of the lower states are calculat’ed in this paper. The molecule is treated as a three-center, two-electron problem. The one-electron Ham.ltonian for t’he electrons in this system includes the electron
FIG.
3 The schematic
representation
of atomic
orbitals
~1
, ~2 , and ~3
ELECTRONIC
STRUCTURE
OF 1)IBORANE
33
repulsion. The Slater-type atomic orbitals are used in this study, while the rffective nuclear caharge of hydrogen is set as 1 and that of boron as 2.15. The latter value was obtained in setting the shielding constant of inner electrons as 0.90, since t,he 2pa eIectron cloud of a negative ion expands more t.han that of a ncut~ral atom. The values of one- and two-center integrals were calculated mainly by tjhcx formula given by Roothaan (14) and by Mulliken’s approximat’ion (15). i4ctual calculation of transition energies is carried out in four steps. E’irst the overlap integrals are calculated. Secondly the ionization potential and thr clew*tronic structure of the pseudo H, atom was evaluated. Then the LCAO-SCl molecular orbital was constructed by minimizing the energy of the ground st:rtc. Finally t,he result is refined by the method of configuration interaction so :I$ to determine the transition energies. V. RESULTS
OF CALCULAT[O,?I
OvERLAP INTEGRALS
The overlap integrals :
were calculated by use of Roothaan’s formula (14). The overlap integral, S, , between Dhe x1 orbit.al of pseudo HZ at,om and the ‘7pn vacant orbital of boron was calculated by t.he following formula and Mulliben’s table (16):
where e is the angle between the B-H bond and the symmetry axis of the 2prr orbital of boron (see Fig. 3). The results are shown in TahIe I. TABLE
I
OVERLAP INTEGRALS"
So = 0.2167 81 = 0.6629 Sn = 0.3515 &SOand Sz are calculated from Root.haan’s proximated formula and Mulliken’s table.
formula,
while S, is calculated
from t.he ap-
34
OGAWA AND HIROTA
THE ELECTRONICSTRUCTUREOF THE Hz GROUP In order to calculate the molecular energy levels, it is necessary to evaluate the wave function and ionization potential of the Hz group. According to the usual treatment of hyperconjugation, the two pseudo AO’s can be constructed from the Is AO’s of the hydrogen atoms as Eq. (1). A neutral H2 group in the valence state has two electrons in the XI orbital, since the lowest r orbital is occupied as mentioned in the preceding chapter. Accordingly the antisymmetrized wave function of the pseudo Hz atom is V = (I/vQ)x~(l)x1(2)
{a(l)@(2)
-
(4)
a(2)/3(I)j.
The wave function of the Hz+ ion, in which an electron in x1 is removed, can be constructed in the same manner as 2J’ = x1(I)&);
Xl(l)P(I).
(5)
The energies of the above states can be written in terms of integrals over the pseudo AO’s : ‘E(H2) = 210 + Jo )
(6)
2E(Hz-t) = lo ,
where lo is the core energy and Jo is the Coulomb integral of the pseudo atom. Therefore, the ionization potential of the Hz group is I,(H,)
= 2E -
‘E = -10
-
Jo.
(7)
The integrals over pseudo atomic orbitals can be reduced to the integrals over the 1s orbitals of hydrogen. IO =
‘?iH
=
PR
=
JO
=
s s
Xl(l)HhCX1(1) dr = [l/(1
a1
j
&)I(~
-
P),
Hh”Ul dr = WH + (a21a1 ad,
alHh"azdr
=
XOWE
+
1xl(l>xl(l)(e2/r12)x1(2)x1(2)
= 2(11
S,)Z {(
+ 2(Ul HhC = T -
-
alall
a21 a1a2>
e2/R1 -
alal) -
4(a1
e2/R2,
(azl alaz),
(S) d7 + (U,Ull a,az) all a1a2)1,
ELECTRONIC
STRUCTURE
OF DIBORANE
:kj
where Wa is the orbital energy of hydrogen 1s AO, T is the kinetic energy operator, Ri is the distance between the electron and the hydrogen core, and (alai 1 utul) and (ai 1aja,) are the Coulomb repulsion integral and the nuclear attraction integral, respectively. The numerical value of Wa is equated to the experimental ionizat’ion potential of hydrogen (13.595 eV). Numerical values of the integrals are calculated by Roothaan’s formula with the effective charge Z = 1. The results are shown in Table II. Thus the ionization potential of the H2 group is
I#L)
= -IO -
Jo
= 6.931 ev.
(W
DETER~IIN~~T~ON OF MOLECULAR ORBITALS Two vacant 3p* orbitals of the borons (~2 and ~3) interact with the x1 orbital of the H, group since they have the same symmetry (see Fig. 3). Suppose the lowest molecular orbital is 41
There are only two 7r electrons determinant is
$1 =
=
ClXl + c&2 + x3). in t’he system,
and then the ground-state
C,4(11
/
SliLtcl
(l/~~)~l(l)~l(a)(a(l)P(2) - &)P(l)J.
In order to determine the coefficients cl and cz st,ate energy, the SW MO mebhod by Roothaan present case, however, since the coefficients cl and tion, the number of the parameters is reduced to state energy was calculated for the various values determined by a graphic method. The ground-st’ate energy is:
-L
00)
(11)
which minimize t*he groundwas often employed. In the c2 are related by normaliznonly one. Thus the grountlof cl and the minimum was
11) + 2Cz4((22 j 22) + (22 I33) + 2(23 123) + a(22 IS){
+ 4c1YJ$?~((11 / 22) + (11 j 23) + 2(12 / 12) + 2(12 1 13)) + 8C1G(ll
1 12)
+ SC,C,3((12 122) + (1” 133) + a(12 j 23)},
(12)
OGAWA
36
AND
HIROTA
TABLE THE
VALUES
II
OF THE INTEGRALS
OF THE Hz GROUP
(a2 1 a,~,) = -7.16+ (a2 I ala,) = -2.949’ = -18.9713 10 zz 12.045 Jo & Calculated b Calculated
by Roothaan’s by Mulliken’s
(eV)
(a,a, I cm) = 17.003a (ala1 ( azuz) = 7.086a (a,~, / am) = 2.610b (a,~, 1cm,) = 0.566b
formula. approximation.
where
Ill = [
HmCXl c/T = WI + q2 I ll),
Xl
+ (31221,
I22 =
w2
+
(1122)
112 =
Sl
Wl
123 =
82
Wz + (1 I 23) + (3 I 23),
+
:a
I 12) + (3 1 la),
(13)
(n ( ij) = /” xi(-e2/R,)xi (ij I kl) =
s
xi(l)xj(l)
wi =
dr, (e”lndxk(2>xt(2)
d7,
-Ip(i),
H,” = T -
En (e'/R,),
where n is any of the bridge hydrogens and the borons. The one- and two-center integrals were again evaluated by Roothaan’s formula, if available, and by Mullken’s approximation. The results are shown in Table III. Detailed descriptions on the evaluation of the integrals are given in the Appendix. W1 corresponds to I, (H2) computed in the preceding section and Wz is equated to the minus of the ionization potential of boron (8.296 eV). The value of El shows the minimum at cl = 0.9990 and c2 = 0.0008. Other molecular orbitals, cpc and & , are calculated from the orthogonality and the normalization conditions, and are shown with their symmetry in parenthesis: (61(W
= 0.9990 x1 + 0.0008 (XZ + x3),
$2 (b3u) =
1.3643 x1 -
1.0285 (x2 + x3),
$3 @,,I =
0.8781 (x2 - x3).
(14)
ELECTItOl”iIC
STRUCTURE
OF l)IBOKAN;E
37
EVALUATION OF THE TR_4NSITION ENERGIES The wave functions for the ground and the lower excited configurations
are
given by J/1(&)
= &(1)&W,
kJ&&)
= (l/v%
(d1(1)9&)
-
(P1(Ws(l)),
VW&u)
= (l/v3
{&(l)$@)
-
+1(0#J3(1) I,
(15)
where the spin fun&ions are omitted. Since $Q and $Q have t,he same symmetry
property,
they interact with each
other and t#heground-stat’e energy is ohtained as the lower root (fi,) of the secular equation :
n-here (17) Kow the energies of each configuration
(&
and EB) and JYz can be expressed in
terms of the atomic integrals as El in Ey. (la). energy
are given
in Table
(2 Ill) = (1 / 22) = (3 j 32) = (2 112) = (3 112) = (1 / 23) = (3 123) = II, I?, II:: ZZ$
a Calculated b Calculated 0 Calculated model. d Calculated
= = = =
(11 11) = 12.04*
-22.601” -21.171” -15.487” -14.982’J -12.624d -7.442d -7.639%
(22 22) = 11.44w (11 22) = 8.385” (22 33) = ci.s01* (11 (12 (13 (12 (12 (22 (11 (12 (23
-52.133 -44.954 -332.201 -17.99i
by Roothaan’s formula. by an approximated method described by Pariser and Parr’s approximation by Mulliken’s
The calculated values of each
IV.
approximaGon.
1‘)) i 22) 22) 12) 13, 23) 23) 23) 23)
= = = = = = = = =
0.77””I 6.573” 5.033(’ 1.423” 3.913~’ 3.207” 2.917” 2.040” 1.127”
it1 the Appendix. of the luliformly
charged
sphere
OGAWA
35
AND
TABLE
HIROTA IV
THE CALCULATED ENERGIES OF EACHSTATEIN REFERENCE TO EI (eV) El = 0 Ez = 1.043 E, = 7.700
E, = -2.848
E, =
3.891
The transition is forbidden from the ground state (E,) to the first excited state (EJ but is allowed to the second excited state (Es). The transition energies are E, -
E, = 6.739 eV,
E3 - E, = 10.548 eV. These results correspond well to the experimental values of 6.9 eV and 9.2 eV. The oscillator strengths for both transitions were calculated by the usual method :
(19) VI.
DISCUSSION
There were many suggestions concerning the bond character in diborane. The idea of protonated double bonds by Yitzer (17) and three-center orbitals by Longuet-Higgins (18) have much to do with the present model. Nulliken (3) showed the structural resemblance of diborane to ethylene and gave a similar molecular orbital description with the present study, though he did not try to evaluate the energies numerically. The calculation of the electronic state of diborane in the present paper was carried out in terms of the hyperconjugation effect, and the obtained transition energies agreed qualitatively with the empirical values. ELECTRONICSPECTRA The assignment of the observed bands in the vacuum ultraviolet region is difficult since not only the a -+ ?r* transition but also the Rydberg transitions occur in this region. Ethylene, which resembles diborane to some extent in its r bond model, has an absorption maximum at 1650 s (7.5 eV), and this band was assigned to the first r -+ T* band (19). The existence of the excited ‘A,, level just above the ground ‘A,, state is one of the results of the present assumption of three-center molecular orbitals. In the ground ‘A,, state, two K electrons lie in the bonding r orbital, $1 ; while in the excited ‘Al, stat’e, one of the g electrons lies in the a orbital 42 , which is antibonding to the bridge hydrogens. The transition between these two states is forbidden by a symmetry selection rule, and consequently the weak band at 1790 ,& is assigned to this transition.
ELECTRONIC
STRUCTURE
OF I)IBORANE
:3!1
In the excited l& state, one of t’he ?r electrons lies in the antihonding a orbital, 4, . The transition to this state is allowed and the strong band at 1350 11 is tentatively assigned to this transition. CALCULATION
In general, ‘,t was difficult and unsuccessful to interpret t)he absorption bands around 2000 A without the consideration of some kind of conjugation in the molecule composed by the zero and first period atoms. The idea of hyperconjugation is useful in discussing the electronic state of molecules especially without any unsaturated bonds and the present calculation adds one more esampk of it. It mny be pointed out that the observed and the calculated energies do not agree quantitatively. In this respect, several assumptions in t’he scheme of calculation and in the evaluation of integrals must he taken into account. For example, if the value of lZs is allowed to decrease to about -17.5 eV, t,he Alc&ted values of the transition energies are 6.97 CV and 9.36 eV, which are it1 good agreement with the observed values. The experimental value of the oscillator strengt$ for the btnd at 1790 _C is approximately 0.002. Those for the bands at 1800 A and 1350 A (7) are roughly estimated to be 0.0005 and 0.3, respectively. The theoretical f value for thrl allowed T - T* transition (0.9) is about three times greater than the obscrvcd value. BOND
CHARWTERISTICS
Three characteristics on the bonding nature of t,he bridge of diboranc were concluded by the determination of potential c&on&ants (2). In the present (*alculation, t’he boron was assumed to have the sp2 orbitals, and two bridge hydrogens were treated as a pseudo atom. In this model a B-HZ-B c bond and a B-HO-B T bond construct the bridgz configuration of diborane. The calculated B-B force constant of 2.721 md/A (2) is a result of these bonds. The valuc~ of t)he overlap integral 82 = 0.3515 is very large as an overlap integral between the noIl-nearest neighbor atoms. Since the calculated value of t)he corresponding int’egral for ethylene is just 0.27, it would be a reasonable assumption to considcl a bond betwren two boron atoms (3). A r bond has little direct,ional property and thus two bridge hydrogens have lit,tle resistivky against the movement of their positions. The calculated small bridge--bond stretching constant and zero bridge-angle deformation constant (2) probably result from this r-bonding nature of t,he bridge. The results of I he calculation of force constants and of electronic energies show the same tendency. CHARGE
DISTRIBUTI~~VS
Concerning the T electron charge distribution, it concentrated mainly in bridge Hz and its negligible fract’ion is distributed on two boron atoms. n-0 at’tempt was
40
OGAWA
AND
HIROTA
made to evaluate the u electron distribution but they are distributed along the B-HZ-B bond. NMR spectra (do), reactivity measurement @I), and nonempirical calculation of electron distribut.ion (4) show that the bridge hydrogen is more negative. Since R electrons concentrate on the bridge hydrogens, they are more negative to the extent that G electrons distribute on them. Thus the acidic character of bridge hydrogen6 as Pitzer suggested (17) is not expected in the present calculation. The authors wish to thank Profess0 st,itute for Solid State Physics of Tokyo traviolet spectrum and discussions.
Saburo Nagakura and Dr. Kozi Kaya of the InUniversity for the measurement of the vacuum ul-
APPENDIX
The integrals were mainly and Mulliken’s approximation
calculated
by use of Roothaan’s
formula
(18)
(14);
the present model is not a typical ?r electron system, there are several integrals which could not be estimated by the suggested formula. The approximations which were used to evaluate such integrals are described here: (1 j 22)-This integral is equated to twice the integral (a 122). In the case where 0 = 90”, the formula for evaluating (a 122) was given in Roothaan’s paper. When 0 = 0”, (a ( 2p r, 2pa) is assumed to be equal to (a 12pa, 2pg) and the value was calculated by Roothaan’s formula. Then the following approximation is used in the present study: Sirxe
(1 j 22) = sin2 0(1 j 22)0_900+ cos2 8(1 122)0_~~. (2 1 12)-Again 110 formula was given in the above literature and the following approximation was used by an analogy of Mulliken’s approximation: (2 1 12) = s1(2 ] 11). RECEIVED:
November 23, 1965 REFERENCES
1. %. 3. 4. 5. 6. 7. 8. 9. 10.
W. N. LIPSCOMB, “Boron Hydrides,” Chap. 2. Benjamin, New York, 1963. T. OGAWA AND T. MIYAZAWA, Spectrochin~. Acta 20, 557 (1964). R. 8. MULLIKEN Chem. Rev. 41, 207 (1947). M. YAMAZAKI, J. Chem. Phys. 27, 1401 (1957). R. HOFFMANNAND W. N. LIPSCOMB,J. Chem. Phys. 37, 2872 (1962). E. BLUM AND G. HERZBERG, J. Phys. Chem. 41, 91 (1937). W. C. PRICE, J. Chem. Phys. 16, 894 (1948). I. HANAZAKI, H. HOSOYA, AND 6. NAGAKURA, Bull. Chem. Sot. Japan 36, 1673 (1963). II. G. WEISS AND I. SHAPIRO, J. Am. Chem. Sot. 81, 6167 (1959). K. HEDBERG AND V. SCHOMAKER,J. Am. Chem. Sot. 73, 1482 (1951); L. S. BARTELL AND B. L. CARROLL,.I. Chem. Phys. 42, 1135 (1965).
ELECTRONIC
STRUCTUIW
OF IIIHOI~ANE
11
11. W. C. H.IMILTON, Proc. Roy. Sot. A236, 395 (1956). 12. L. P.\~LING, “The Nature of the Chemical Bond.” Cornell Univ. Press, New York, 1960. 13. C. A. COULSON, “Valence,” p. 360. Oxford IJniv. Press, London and New York, 1961. 14. C. C. J. ROOTH.~AN, J. C’hem. Phys. 19, 1445 (1951). 15. R. 8. MULLIEEN, J. Chim. Phys. 46, 500 (1919). 16. R. S. MULLIKEN, C. A. RIEKE, D. ORI,OFF, .\sI) II. OIZLOFF, J. Chem. Phys. 17, 1248 (1949). 27. K. S. PITZER, J. Bm. Chem. Sot. 67, 112ti (1945). 18. H. C. LONGUET-HIGGINS, J. Chem. Phys. 46, 275 (1!149). 1.9. H. H. JI\FFI%AND M. ORCHIN, “ Theory and Applicat,iorrs of Ultraviolet Spectroscopy.” Wiley, New York, 1962. 30. R. A. OGG, J. Chem. Phys., 22, 1933 (1954). 31. A. BORG, J. Am. Chem. Sot. 69, 747 (1947).
OF MOLECULAR
SPECTROSCOPY
21, 29-41
(1966)
Electronic Structure of Di borane and Hyperconjugation Effect TEIICHIRO Department
OGAWA AND Kozo
sf Chemistry, Faculty of Scieme,
Osaka
HIROT_~
lJniz)ersity,
Toyonaka,
Osaka, Japan
The ultraviolet
spectra of dihorane are re-examined and its electronic strucby the method of LCAO SCF. The hyperconjugat,ion effect of the bridge hydrogens is taken into consideration and t,he system is treated ns :t three-center, two-electron problem. The int,egrals are mainly calculated nonempirically but some of them are evaluated by making some approximation. The transit.ion energies are evaluated to be 6.7 (forbidden) and 10.5 e\ (allowed), and they correspond web to the experimental values of 6.9 (weak) and 9.2 eV (strong), respectively. The hyperconjngation effect is proved to play an important role in siirh a bridge compound. The nature of the bridge bond is discussed.
1ure is calculated
I. INTRODUCTIOS
The peculiar structures of boron hydrides have invited many investigators in bhis field. Diborane, the simplest of these hydrides, was concluded to he in a bridged st’ructure and two hydrogens of diborane lie between two boron atoms. TTarious interpretations were proposed to elucidate the structure of this molecule (I). Ogawa and Miyazawa (2) have recently calculated the normal vihrations of diborane and showed several characteristics of the bond in the bridge. Some type of bond was concluded to exist between two boron atoms through the analysis of the potential function, but’ no detailed explanation of bond character n-as proposed. Mulliken (S) discussed the MO electronic configuration for both ethane-like and bridge structure, and Yamazaki (4) calculated the self-consistent field molecular orbitals in LCAO forms for the ground state of diborane by Roothaan’s procedure. Hoffmann and Lipscomb (5) investigated the whole series of boron hydrides in t’erms of the LCAO treatment. The electronic absorption spectra of diborartc were measured by Rlum et al. (6) and Price (7). Three bands were found at lS20 i (6.S eT’ weak), 13.50 8 (9.3 eV strong), and 1200 A (10.3 eT_. strong). Little attempts, however, have been made to interpret these bands in terms of the LCAO treatment. Nethods of calculation have been developed on the electronic structure of various molecules quite satisfnctorilp. Many complex molecules, especially 29
30
OGAWA
AND
HIROTA
unsaturated hydrocarbons, have been made the objects of study, and their electronic spectra have been elucidated in terms of molecular energy levels. Hanazaki, Hosoya, and Nagakura (8) studied the electronic structure of the tbutyl cation and showed the importance of hyperconjugation effect in the molecules without any unsaturated bonds. Diborane also has no typical double bond, but two bridge hydrogens are situated as suitable for hyperconjugation with two vacant 2p7r orbitals of borons. In the present study, the spectrum was re-examined and the electronic structure of diborane is discussed in terms of the self-consistent field molecular orbitals in LCAO forms and the hyperconjugation effects of bridge hydrogens with two borons. The object of the present study is to assign the observed absorption bands to corresponding transitions of molecular energy levels and to clarify the nature of the bridge bond of this molecule. II. EXPERIMENTAL
Diborane was synthesized by adding sodium borohydride slowly to cold concent,rated sulfuric acid (9). In order to separate diborane from hydrogen, hydrogen sulfide, and sulfur dioxide which were reported to be produced as byproduct’s, the products were cooled down with liquid nitrogen and were evacuated. Then the liquid nitrogen was replaced by dry ice-methanol and the evolved gas was collected in a Toepler pump. Hydrogen sulfide was found in the sample to be 0.8 per cent and a trace of sulfur dioxide was found by the mass-spectrometric analysis. The ultraviolet spectra were measured by a Hitachi EPS-2 spectrophotometer in the 3400-2200 A region, by a Cary 14 PM spectrophotometer in the 22001900 8 region, and by a vacuum ultraviolet spectrometer at the Institute for Solid State Physics in the 2050-1550 i region. III.
EXPERIMENTAL
RESULTS
There was no absorption band in the 3400-1900 _®ion. The observed spectrum in the vacuum ultraviolet region is shown in Fig. 1. An absorption band was located at 1790 1 and its extinction coefficient was about 90. The increase of absorbance was found from 1650 i to the shorter wavelength region, but no absorption peak would be observed. The absorption band at 1790 A is assigned to a band forbidden ,by a symmetry selection rule because of its weak intensity. The band at 1350 A found by Price is perhaps the first allowed transition, and this band is tentatively assigned to the T -+ n* transition. IV. GENERAL
OUTLINE
OF THE
CALCULATION
The structure and the molecular dimensions of diborane were determined by the electron diffraction study (10) and are shown in Fig. 2. TWO bridge hydrogens lie in a plane normal to the plane of four terminal hydrogens. The bonds to the four terminal hydrogens are considered essentially single bonds and are as-
ELECTRONIC
FIG. 1. The
electronic
FIG. 2. The structure
spectrllm
of diborane.
STRUCTURE
of diborane
31
OF DIBORANE
in the vac,lum
ultraviolet
region
n = 1.19G d, 6 = 1.339 d, I, = 1.775 A, and a =
119.0”.
sumed to be sufficiently localized so that t’he bridge bonds are separated from t’he remaining part of the molecule (11). The terminal ( HBH angle is 122” and the B-B distance (1.775 A) is approximately equal to t,he sum (1.76 8) of Pauling’s tetrahedral covalent radii (1.2). Accordingly each boron atom is regarded to have the sp2 hybrid orbitals to the other boron atom as well as to t’wo t’erminal hydrogens and to have a vacant 2pprr
32
OGAWA
AND
HIROTA
orbital normal to the hybrid orbitals. Two bridge hydrogen atoms are treated as a united atom which has a pseudo u orbital x0 and a pseudo r orbital x1 :
x0 = M2U + SO>l’/2(UI + 4, Xl =
l/Lw
-
So>l’/2(al
-
(1)
f-321,
where al and aa are 1s orbitals of the bridge hydrogens. Similar assumptions are often adopted in the treatment of hyperconjugation of the methyl group (8, IS). Two 2pa orbitals of the borons (xZ and x3) and the x1 orbital of t’he bridge hydrogens interact with each other and make a T molecular orbital like that of ally1 radical (see Fig. 3). Resemblance in molecular structure of t’his compound to ethylene may allow the present treatment. Two .s@’ orbitals of borons make a B-B bond, with which the ~0 orbital of the bridge hydrogens will interact, and they make bonding and antibonding u orbitals. There are four electrons in these bridge bonds; two come from borons and two from bridge hydrogens. These four electrons are situated in the lowest and the second lowest orbitals. The lowest orbital must be the lowest bonding u orbit,al, B-HZ-B. The second lowest orbital is concluded not to be the second 0 orbital but to be a bonding a orbital, since the result of Hamilton (11) and the result of the nonempirical calculation by Yamazaki (4) show that the a-like orbital 47 (in Yamazaki’s notation) is very stable energetically. That is to say, in the bridge of diborane two electrons lie in the v orbital and the remaining two electrons in the P orbital; the situation is just like that of ethylene. In order to interpret the ultraviolet spectra, the energies of these 7relectrons are to be calculated. Only the energies of the lower states are calculat’ed in this paper. The molecule is treated as a three-center, two-electron problem. The one-electron Ham.ltonian for t’he electrons in this system includes the electron
FIG.
3 The schematic
representation
of atomic
orbitals
~1
, ~2 , and ~3
ELECTRONIC
STRUCTURE
OF 1)IBORANE
33
repulsion. The Slater-type atomic orbitals are used in this study, while the rffective nuclear caharge of hydrogen is set as 1 and that of boron as 2.15. The latter value was obtained in setting the shielding constant of inner electrons as 0.90, since t,he 2pa eIectron cloud of a negative ion expands more t.han that of a ncut~ral atom. The values of one- and two-center integrals were calculated mainly by tjhcx formula given by Roothaan (14) and by Mulliken’s approximat’ion (15). i4ctual calculation of transition energies is carried out in four steps. E’irst the overlap integrals are calculated. Secondly the ionization potential and thr clew*tronic structure of the pseudo H, atom was evaluated. Then the LCAO-SCl molecular orbital was constructed by minimizing the energy of the ground st:rtc. Finally t,he result is refined by the method of configuration interaction so :I$ to determine the transition energies. V. RESULTS
OF CALCULAT[O,?I
OvERLAP INTEGRALS
The overlap integrals :
were calculated by use of Roothaan’s formula (14). The overlap integral, S, , between Dhe x1 orbit.al of pseudo HZ at,om and the ‘7pn vacant orbital of boron was calculated by t.he following formula and Mulliben’s table (16):
where e is the angle between the B-H bond and the symmetry axis of the 2prr orbital of boron (see Fig. 3). The results are shown in TahIe I. TABLE
I
OVERLAP INTEGRALS"
So = 0.2167 81 = 0.6629 Sn = 0.3515 &SOand Sz are calculated from Root.haan’s proximated formula and Mulliken’s table.
formula,
while S, is calculated
from t.he ap-
34
OGAWA AND HIROTA
THE ELECTRONICSTRUCTUREOF THE Hz GROUP In order to calculate the molecular energy levels, it is necessary to evaluate the wave function and ionization potential of the Hz group. According to the usual treatment of hyperconjugation, the two pseudo AO’s can be constructed from the Is AO’s of the hydrogen atoms as Eq. (1). A neutral H2 group in the valence state has two electrons in the XI orbital, since the lowest r orbital is occupied as mentioned in the preceding chapter. Accordingly the antisymmetrized wave function of the pseudo Hz atom is V = (I/vQ)x~(l)x1(2)
{a(l)@(2)
-
(4)
a(2)/3(I)j.
The wave function of the Hz+ ion, in which an electron in x1 is removed, can be constructed in the same manner as 2J’ = x1(I)&);
Xl(l)P(I).
(5)
The energies of the above states can be written in terms of integrals over the pseudo AO’s : ‘E(H2) = 210 + Jo )
(6)
2E(Hz-t) = lo ,
where lo is the core energy and Jo is the Coulomb integral of the pseudo atom. Therefore, the ionization potential of the Hz group is I,(H,)
= 2E -
‘E = -10
-
Jo.
(7)
The integrals over pseudo atomic orbitals can be reduced to the integrals over the 1s orbitals of hydrogen. IO =
‘?iH
=
PR
=
JO
=
s s
Xl(l)HhCX1(1) dr = [l/(1
a1
j
&)I(~
-
P),
Hh”Ul dr = WH + (a21a1 ad,
alHh"azdr
=
XOWE
+
1xl(l>xl(l)(e2/r12)x1(2)x1(2)
= 2(11
S,)Z {(
+ 2(Ul HhC = T -
-
alall
a21 a1a2>
e2/R1 -
alal) -
4(a1
e2/R2,
(azl alaz),
(S) d7 + (U,Ull a,az) all a1a2)1,
ELECTRONIC
STRUCTURE
OF DIBORANE
:kj
where Wa is the orbital energy of hydrogen 1s AO, T is the kinetic energy operator, Ri is the distance between the electron and the hydrogen core, and (alai 1 utul) and (ai 1aja,) are the Coulomb repulsion integral and the nuclear attraction integral, respectively. The numerical value of Wa is equated to the experimental ionizat’ion potential of hydrogen (13.595 eV). Numerical values of the integrals are calculated by Roothaan’s formula with the effective charge Z = 1. The results are shown in Table II. Thus the ionization potential of the H2 group is
I#L)
= -IO -
Jo
= 6.931 ev.
(W
DETER~IIN~~T~ON OF MOLECULAR ORBITALS Two vacant 3p* orbitals of the borons (~2 and ~3) interact with the x1 orbital of the H, group since they have the same symmetry (see Fig. 3). Suppose the lowest molecular orbital is 41
There are only two 7r electrons determinant is
$1 =
=
ClXl + c&2 + x3). in t’he system,
and then the ground-state
C,4(11
/
SliLtcl
(l/~~)~l(l)~l(a)(a(l)P(2) - &)P(l)J.
In order to determine the coefficients cl and cz st,ate energy, the SW MO mebhod by Roothaan present case, however, since the coefficients cl and tion, the number of the parameters is reduced to state energy was calculated for the various values determined by a graphic method. The ground-st’ate energy is:
-L
00)
(11)
which minimize t*he groundwas often employed. In the c2 are related by normaliznonly one. Thus the grountlof cl and the minimum was
11) + 2Cz4((22 j 22) + (22 I33) + 2(23 123) + a(22 IS){
+ 4c1YJ$?~((11 / 22) + (11 j 23) + 2(12 / 12) + 2(12 1 13)) + 8C1G(ll
1 12)
+ SC,C,3((12 122) + (1” 133) + a(12 j 23)},
(12)
OGAWA
36
AND
HIROTA
TABLE THE
VALUES
II
OF THE INTEGRALS
OF THE Hz GROUP
(a2 1 a,~,) = -7.16+ (a2 I ala,) = -2.949’ = -18.9713 10 zz 12.045 Jo & Calculated b Calculated
by Roothaan’s by Mulliken’s
(eV)
(a,a, I cm) = 17.003a (ala1 ( azuz) = 7.086a (a,~, / am) = 2.610b (a,~, 1cm,) = 0.566b
formula. approximation.
where
Ill = [
HmCXl c/T = WI + q2 I ll),
Xl
+ (31221,
I22 =
w2
+
(1122)
112 =
Sl
Wl
123 =
82
Wz + (1 I 23) + (3 I 23),
+
:a
I 12) + (3 1 la),
(13)
(n ( ij) = /” xi(-e2/R,)xi (ij I kl) =
s
xi(l)xj(l)
wi =
dr, (e”lndxk(2>xt(2)
d7,
-Ip(i),
H,” = T -
En (e'/R,),
where n is any of the bridge hydrogens and the borons. The one- and two-center integrals were again evaluated by Roothaan’s formula, if available, and by Mullken’s approximation. The results are shown in Table III. Detailed descriptions on the evaluation of the integrals are given in the Appendix. W1 corresponds to I, (H2) computed in the preceding section and Wz is equated to the minus of the ionization potential of boron (8.296 eV). The value of El shows the minimum at cl = 0.9990 and c2 = 0.0008. Other molecular orbitals, cpc and & , are calculated from the orthogonality and the normalization conditions, and are shown with their symmetry in parenthesis: (61(W
= 0.9990 x1 + 0.0008 (XZ + x3),
$2 (b3u) =
1.3643 x1 -
1.0285 (x2 + x3),
$3 @,,I =
0.8781 (x2 - x3).
(14)
ELECTItOl”iIC
STRUCTURE
OF l)IBOKAN;E
37
EVALUATION OF THE TR_4NSITION ENERGIES The wave functions for the ground and the lower excited configurations
are
given by J/1(&)
= &(1)&W,
kJ&&)
= (l/v%
(d1(1)9&)
-
(P1(Ws(l)),
VW&u)
= (l/v3
{&(l)$@)
-
+1(0#J3(1) I,
(15)
where the spin fun&ions are omitted. Since $Q and $Q have t,he same symmetry
property,
they interact with each
other and t#heground-stat’e energy is ohtained as the lower root (fi,) of the secular equation :
n-here (17) Kow the energies of each configuration
(&
and EB) and JYz can be expressed in
terms of the atomic integrals as El in Ey. (la). energy
are given
in Table
(2 Ill) = (1 / 22) = (3 j 32) = (2 112) = (3 112) = (1 / 23) = (3 123) = II, I?, II:: ZZ$
a Calculated b Calculated 0 Calculated model. d Calculated
= = = =
(11 11) = 12.04*
-22.601” -21.171” -15.487” -14.982’J -12.624d -7.442d -7.639%
(22 22) = 11.44w (11 22) = 8.385” (22 33) = ci.s01* (11 (12 (13 (12 (12 (22 (11 (12 (23
-52.133 -44.954 -332.201 -17.99i
by Roothaan’s formula. by an approximated method described by Pariser and Parr’s approximation by Mulliken’s
The calculated values of each
IV.
approximaGon.
1‘)) i 22) 22) 12) 13, 23) 23) 23) 23)
= = = = = = = = =
0.77””I 6.573” 5.033(’ 1.423” 3.913~’ 3.207” 2.917” 2.040” 1.127”
it1 the Appendix. of the luliformly
charged
sphere
OGAWA
35
AND
TABLE
HIROTA IV
THE CALCULATED ENERGIES OF EACHSTATEIN REFERENCE TO EI (eV) El = 0 Ez = 1.043 E, = 7.700
E, = -2.848
E, =
3.891
The transition is forbidden from the ground state (E,) to the first excited state (EJ but is allowed to the second excited state (Es). The transition energies are E, -
E, = 6.739 eV,
E3 - E, = 10.548 eV. These results correspond well to the experimental values of 6.9 eV and 9.2 eV. The oscillator strengths for both transitions were calculated by the usual method :
(19) VI.
DISCUSSION
There were many suggestions concerning the bond character in diborane. The idea of protonated double bonds by Yitzer (17) and three-center orbitals by Longuet-Higgins (18) have much to do with the present model. Nulliken (3) showed the structural resemblance of diborane to ethylene and gave a similar molecular orbital description with the present study, though he did not try to evaluate the energies numerically. The calculation of the electronic state of diborane in the present paper was carried out in terms of the hyperconjugation effect, and the obtained transition energies agreed qualitatively with the empirical values. ELECTRONICSPECTRA The assignment of the observed bands in the vacuum ultraviolet region is difficult since not only the a -+ ?r* transition but also the Rydberg transitions occur in this region. Ethylene, which resembles diborane to some extent in its r bond model, has an absorption maximum at 1650 s (7.5 eV), and this band was assigned to the first r -+ T* band (19). The existence of the excited ‘A,, level just above the ground ‘A,, state is one of the results of the present assumption of three-center molecular orbitals. In the ground ‘A,, state, two K electrons lie in the bonding r orbital, $1 ; while in the excited ‘Al, stat’e, one of the g electrons lies in the a orbital 42 , which is antibonding to the bridge hydrogens. The transition between these two states is forbidden by a symmetry selection rule, and consequently the weak band at 1790 ,& is assigned to this transition.
ELECTRONIC
STRUCTURE
OF I)IBORANE
:3!1
In the excited l& state, one of t’he ?r electrons lies in the antihonding a orbital, 4, . The transition to this state is allowed and the strong band at 1350 11 is tentatively assigned to this transition. CALCULATION
In general, ‘,t was difficult and unsuccessful to interpret t)he absorption bands around 2000 A without the consideration of some kind of conjugation in the molecule composed by the zero and first period atoms. The idea of hyperconjugation is useful in discussing the electronic state of molecules especially without any unsaturated bonds and the present calculation adds one more esampk of it. It mny be pointed out that the observed and the calculated energies do not agree quantitatively. In this respect, several assumptions in t’he scheme of calculation and in the evaluation of integrals must he taken into account. For example, if the value of lZs is allowed to decrease to about -17.5 eV, t,he Alc&ted values of the transition energies are 6.97 CV and 9.36 eV, which are it1 good agreement with the observed values. The experimental value of the oscillator strengt$ for the btnd at 1790 _C is approximately 0.002. Those for the bands at 1800 A and 1350 A (7) are roughly estimated to be 0.0005 and 0.3, respectively. The theoretical f value for thrl allowed T - T* transition (0.9) is about three times greater than the obscrvcd value. BOND
CHARWTERISTICS
Three characteristics on the bonding nature of t,he bridge of diboranc were concluded by the determination of potential c&on&ants (2). In the present (*alculation, t’he boron was assumed to have the sp2 orbitals, and two bridge hydrogens were treated as a pseudo atom. In this model a B-HZ-B c bond and a B-HO-B T bond construct the bridgz configuration of diborane. The calculated B-B force constant of 2.721 md/A (2) is a result of these bonds. The valuc~ of t)he overlap integral 82 = 0.3515 is very large as an overlap integral between the noIl-nearest neighbor atoms. Since the calculated value of t)he corresponding int’egral for ethylene is just 0.27, it would be a reasonable assumption to considcl a bond betwren two boron atoms (3). A r bond has little direct,ional property and thus two bridge hydrogens have lit,tle resistivky against the movement of their positions. The calculated small bridge--bond stretching constant and zero bridge-angle deformation constant (2) probably result from this r-bonding nature of t,he bridge. The results of I he calculation of force constants and of electronic energies show the same tendency. CHARGE
DISTRIBUTI~~VS
Concerning the T electron charge distribution, it concentrated mainly in bridge Hz and its negligible fract’ion is distributed on two boron atoms. n-0 at’tempt was
40
OGAWA
AND
HIROTA
made to evaluate the u electron distribution but they are distributed along the B-HZ-B bond. NMR spectra (do), reactivity measurement @I), and nonempirical calculation of electron distribut.ion (4) show that the bridge hydrogen is more negative. Since R electrons concentrate on the bridge hydrogens, they are more negative to the extent that G electrons distribute on them. Thus the acidic character of bridge hydrogen6 as Pitzer suggested (17) is not expected in the present calculation. The authors wish to thank Profess0 st,itute for Solid State Physics of Tokyo traviolet spectrum and discussions.
Saburo Nagakura and Dr. Kozi Kaya of the InUniversity for the measurement of the vacuum ul-
APPENDIX
The integrals were mainly and Mulliken’s approximation
calculated
by use of Roothaan’s
formula
(18)
(14);
the present model is not a typical ?r electron system, there are several integrals which could not be estimated by the suggested formula. The approximations which were used to evaluate such integrals are described here: (1 j 22)-This integral is equated to twice the integral (a 122). In the case where 0 = 90”, the formula for evaluating (a 122) was given in Roothaan’s paper. When 0 = 0”, (a ( 2p r, 2pa) is assumed to be equal to (a 12pa, 2pg) and the value was calculated by Roothaan’s formula. Then the following approximation is used in the present study: Sirxe
(1 j 22) = sin2 0(1 j 22)0_900+ cos2 8(1 122)0_~~. (2 1 12)-Again 110 formula was given in the above literature and the following approximation was used by an analogy of Mulliken’s approximation: (2 1 12) = s1(2 ] 11). RECEIVED:
November 23, 1965 REFERENCES
1. %. 3. 4. 5. 6. 7. 8. 9. 10.
W. N. LIPSCOMB, “Boron Hydrides,” Chap. 2. Benjamin, New York, 1963. T. OGAWA AND T. MIYAZAWA, Spectrochin~. Acta 20, 557 (1964). R. 8. MULLIKEN Chem. Rev. 41, 207 (1947). M. YAMAZAKI, J. Chem. Phys. 27, 1401 (1957). R. HOFFMANNAND W. N. LIPSCOMB,J. Chem. Phys. 37, 2872 (1962). E. BLUM AND G. HERZBERG, J. Phys. Chem. 41, 91 (1937). W. C. PRICE, J. Chem. Phys. 16, 894 (1948). I. HANAZAKI, H. HOSOYA, AND 6. NAGAKURA, Bull. Chem. Sot. Japan 36, 1673 (1963). II. G. WEISS AND I. SHAPIRO, J. Am. Chem. Sot. 81, 6167 (1959). K. HEDBERG AND V. SCHOMAKER,J. Am. Chem. Sot. 73, 1482 (1951); L. S. BARTELL AND B. L. CARROLL,.I. Chem. Phys. 42, 1135 (1965).
ELECTRONIC
STRUCTUIW
OF IIIHOI~ANE
11
11. W. C. H.IMILTON, Proc. Roy. Sot. A236, 395 (1956). 12. L. P.\~LING, “The Nature of the Chemical Bond.” Cornell Univ. Press, New York, 1960. 13. C. A. COULSON, “Valence,” p. 360. Oxford IJniv. Press, London and New York, 1961. 14. C. C. J. ROOTH.~AN, J. C’hem. Phys. 19, 1445 (1951). 15. R. 8. MULLIEEN, J. Chim. Phys. 46, 500 (1919). 16. R. S. MULLIKEN, C. A. RIEKE, D. ORI,OFF, .\sI) II. OIZLOFF, J. Chem. Phys. 17, 1248 (1949). 27. K. S. PITZER, J. Bm. Chem. Sot. 67, 112ti (1945). 18. H. C. LONGUET-HIGGINS, J. Chem. Phys. 46, 275 (1!149). 1.9. H. H. JI\FFI%AND M. ORCHIN, “ Theory and Applicat,iorrs of Ultraviolet Spectroscopy.” Wiley, New York, 1962. 30. R. A. OGG, J. Chem. Phys., 22, 1933 (1954). 31. A. BORG, J. Am. Chem. Sot. 69, 747 (1947).