An Alternate Method of Determining styx Values and Their Utility Stanley C. Grenda University of Nevada, Las Vegas, Las Vegas, NV 89154
In 1954, Eberhardt, Crawford, and Lipscomb' proposed the "styx" concept for the valence structure of the then known boron hydrides having the general formula BPHv The letters styx were defined as follows: s = number of hydrogen bridges, B-H-B, t = number of boron-boron three-center bonds, B
B.
B
The method proposed in this article simplifies and expands on that of Lipscomb by allowing for both the calculation and use of fractional and negative styx values. The method is based on the general formulaBpH: and the following assumptions: 1. AU bands are two- or three-center,two-electron honds. 2. Every boron is bonded to at least one hydrogen by a two-electron,
y = number of boron-boron single bonds, B-B,
and x = number of extra H atoms on B-H; i.e., BHz groups. The styx values were determined based on the following fundamental assumptions: 1. The 1s orbital of hydrogen and the four spa orbitals of boron are
used. 2. Of the structures considered, every boron foms at least one BH bond. 3. Each B-H-B bridge bond is regarded as a filled, three-center, localized bonding orbital requiring the hydrogen orbital and one hybrid orbital from each boron. 4. The orbitals and electrons of any particular boron are allotted so as to satisfv first the reauirements of the external B--H sinele bonds and'tbe bridge B ~ H - B bonds. The remaining orbit& and electrons contribute to framework molecular orbitals, namely those orbitals and electrons used in the formation of boron twa- and tbree-center bonds, respectively, designated by the letters t and y above. These assumptions led to the following equations: Orbital balance,
two-center bond (B-H). 3. Every boron contributes all four orbitals to bonding. 4. Only nonnegative sty%values for known boron hydrides are acceptable. 5. For BHE, fragments, fractional and negative styx values are acceptable. Equation3 9-16 are a direct result of the aforementioned assumptions. The total number of bonds possible is equal to the total number of valence electrons available divided by two. Total bond number = (3p + q - c)/2
(9)
Assumption 2 yields eq 10, Remaining bond number = (3p + q - e)/2 - p = Lo + q - c)/2 (10) In eq 10, the term -p represents the number of B-H Assumption 3 yields eq 11.
bonds.
Total number of orbitals available = 4p + q Remaining orbitals available = 4p + q - 2p = 2p
(11)
+q
(12)
In eq 12 the term -2p represents the number of orbitals used in forming p (B-H) bonds.
Electron accounting, The authors added the need of a "hydrogen balance" and a requirement that t and y be nonnegative, which resulted in these additional relationships:
'Using eqs 10 and 12, one is able t o generate eq 13. Equation 13 allows us to calculate the number of two-center and threecenter honds left in the molecule or ion, where a = the number of two center bonds and @ q - c ) / 2 - a = the number of three center bonds. Recalling that s and t are three-center bonds and that y and x are two-center honds, we get two additional relationships:
+
s+t=@+q-c)/2-a Sometime later Lipscomb expanded on these equations t o include the anionic boron hydrides having the general formula BpH;+ +. This expansion led t o slight changes resulting in eqs 6-8: ' Electron accounting,
(14)
Bond and Olbltal Count from Eps 9-13
Table 1.
Total Remain-
Remain- Twc-Ceme? Three4enterb Bond ing Bond ecule Number Number Orbitals Orbitals Number Number
Hydrogen atom balance, atr=q+c
'
(7)
Eberhardt, W. H.: Crawford, B. L.. Jr.; Lipscomb. W. N. J. Chem. Phys. 1954,22,989. Lipscomb. W. N. J. Phys. Chem. 1958, 62,381.
Mi-
Bond
BrHD B3Hs
12 8 14
BsHs
ing Bond Total
7 5 8
29 18 32
19 12 20
2 3 4
5 2 4
'TWO center Wnd nmber = y + x. aThrea center Wndnmbet = r + f
Volume 66
Number 8 August 1989
639
Table 2.
sh~Values Y
x
4
2
2
3 2
3 4
0
Molecule
s
t
&He
0 1
2
1
N w t i v e valveof f is unaccepnble, andtherefore mir mwof valuss Is unacceptable.
enables one to quickly check his or her calculations for a possible error. If s t y x = @ q - c)/2, an error in the styx values exists, and the values need to be recomputed. If the eaualitv holds. there is no euarantee that the d u e s are correct, so eqs 14 and 15should-be used aschecks for thestyx values. Usina eas 9-16. one can calculate the fractional and negative values-of the'simple fragments found in Tahle 3. Some of these fragments have been observed in the mass spectra of the follo&ng boron hydrides-B2H6, B4Hlo,BsH9, and BsH11-by Bracp et al.3 The stvx values of these fraements, just as for k n % n horon hydrides, possess an additive and subtractive property. For example, if one adds BH+ (02 - 1.50) to BH- (00 - 1.50), one gets B2H2 (0200), the value obtained by Lipscomh. Application of eqs 9-15 to BeHs is demonstrated here:
+ + +
Tabla 3. styx Values of BH; Fragments Fragment
s
t
BHt BH BH-
0 0 0 2
2
BH: BHn BHE BHs BH3 BHh BH, BHa BH;
1
0 2 1
0 2 1
0
Bond number = [3(6)+ 8 - (-2)]/2 = 14
-1.5 0 1.5
o
o
0 0 0 0 0 0 0 0
0.5 1 -0.5 0 0.5 -1
-0.5 0
0 0 0
Journal of Chemical Education
=8
+
Total number of orbitals available = 4(6) 8 = 32 Remaining orbitals available = 2(6) + 8 = 20
-1
2a
0
+ 3(8 - a) = 20
1
0 1 2 1 2 3
Using these equations and the hydrides listed inTable 1,one is able to calculate the values listed in Tables 1and 2. The styx values listed in Tahle 2 agree exactly with those of Lipscomh e t al. The use of eq 16,
640
Remaining bond number = [6 + 8 - (-2)]/2
x
Y
1 0
+
With two hydrogens remaining unbonded, the maximum valuex can have is 2. Possible values of x are, 2,1,O yielding t values of O,1,2, respectively. If x = 2 then s = 0 and t = 4, etc. This same reasoning is applied to all boron hydride species, he they molecules, ions, or fragments. Bragg. J. K.: McCariy, L. V.; Norton, F. J. J. Am. Chem. Soc. 1951,
73,2134.
