THE CONCEPT OF H~DRIDISATION

of first and second order hybridisation isundertake'n, and

it is concluded that in no sense can these concepts be said

to eXElain molecular bond angles or dipole moments.

Because of the i~~ense complexity of even the simplest

molecular systems, the theory of the chemical bond, has, of

necessity, been developed along semi-empirical lines. This

approach was initiated by PaUling(l) whose basis for directed

valence has its origin in the electron-pair formulation of

chemical binding. He postulated that: ~Of two eigenfunctions

with the same radial dependence, the one with the larger'f'1!.C, ;1..g

value in the bond direction will giveAto a stronger bond,

and for a given eigenfunction, the bond will tend to fornl

in the direction with the largest value of the eigenfunction.~

Apart from slight modifications (e.g. the suggested replacement

of "eigenfunction" by' overlap integral"), this postulate has

been unquestioningly accepted as the basis of stereochemical

theory; it has been widely developed in the discussion of

molecular shapes and is generally taken as solving, at least

,/<

- 2 -

in principle, the main problem of molecular structure.

We believe that the success of the theory is fortuitous.

However we do not deny that it serves a very useful purpose,

from a utilitarian point of view, in fostering an intuitive

approach to this problem so that chemists can rationalise about

bomd angles without having any fundamenta~idea of the factors

governing their magnitude. IIowever this usefulness has become

marred by indiscriminate extensionf to the discussion of many

other molecular phenomena; for with the passage of time, chemists

have sometimes lost sight of the basic principles of quantum

mechanics, and the formalism that has evolved is no longer

internally consistent.

First Order Hybridisation.

It is convenient to treat an n-electron atom as if each

individual electron were in a stationary state represented by

a one-electron wave function. The wave function of the atom

is then an antis~[@etrised product of these functions, i.e.

it may be written in the form of an n-th. order determinant,

as first shown by Dirac, ~

- 3 -

<j)~(l)$SS2(1) .M ...... $k(l)

'I\r =

I $~(2)$~ (2) ••••...• ¢~( 2)·

· ·· • •• · •·¢~(n)

$s ¢s2(n) •..••••• ken)

where s denotes the nature of the spin factor of the orbitals,

and k is an integer not less than n and not greater than 2n.

This type of wave function is consistent with the fact that

electrons are indistinguishable, and has been most successfully

used in the systematisation of atomic structure.

It is only within the limits of this approximation

that hybridisation is meaningful. To simplify the discussion

we will aSS1.L1Jlethat the valence state of an ayom can be

represented by a single determinant (because the concept of

the valence state cannot be divorced in general from that

of second order hybridisation). If we take the 4-valent

state of carbon we can divide the electrons into the closed

shells with paired spins and the four valence electrons whose

spins are random (in the valence state). We can, following

convention, ignore the closed shells and consider only the

fourth order determinant; det.I§(l), p (1), p (1), p (1) I,'. ··x -y -zIn 4bb"~\h..;..l:-;:D"" otd"t tl..i? fl-f"5t +O~ III ~t"lt"'e".

where, tho ngt~tiQn ig ob~ioug. 80nsidering the simplest case,

- 4 -

digonal hybridisation, we can readily transform the

determinant ~y operations on suitable multiples of columns

(1) and (2)] into a new fourth order determinant

which leaves the total W unchanged. We have not altered the

atom in any significant way, but have only re-allocated the

electrons to a different set of one-electron functions, which

when made antisymmetric are identical with the original set.

The question has been asked: Why can we take linear

combinations of § and p. functions when they are not degenerate?

This is a meaningless question and arises from a too literal

acceptance of the orbital picture. The orbitals themselves

are not solutions of the Schr~dinger equation for the atom

in question: they are only easilyvisualisable trial functions

which when antis~mmetrised lead to a reasonable description

of the atom's chemical properties. As such the only restriction

on combining them is that each column must correspond to

exactly one electron, and the amount of ~!electricity and

the amount of p electricity shall both remain constant, (i.e.

no promotion). The functions of each column must also,of

course,be mutually orthogonal.

---------------------------

- 5 -

Second Order Hybridisation.

This term is used to denote the superposition of two or

more orbital configurations in the formulation of the valence

state. Consider the configuration sp. There are four

spectroscopic sub-states or multiplets whose wave functions

are(2):-

w ( 3p, 1, 1) = det. I~(1), P ( 1) I .

w(3p, 1,-1) = det. l~(l), p(l)l.

w(3p, 1, 0) =(l)~[c1et.I~(l), p(l)1 + det. I~(l), p(l)IJ~~.

1jf(lp,1, 0) =(~)Tdet.I~(l), p(l)1 - det. I~(l), p(l)IJ_.

The first two functions represent states with spins of 1 and

-1 respectively, but the last two have zero expectation value

of the spin. Already the orbital theory is beginning to

lose itts simple pictorial character. For in say w(3p, 1, 0)',,, ~,\. 0.\'.1" ••1

we cannot say ~~ the electron with spin +~ is; ao far ~

orbitaA~ ~~9 conocrned. we cannot actually say that the s

electron has a definite spin for if we observe it a large

nurnber of times we wil~as many results +i as -to M:Offitt(2)~

has shown that the wave function of the valence state of any

atom can be written as

- 6 -

\1; = Z.'v 1 a.\ir f2s+1l"i' L, Lz,

s ).z

the sur@lation being strictly over the whole spectrwli of the

atom, the notation also implying an integration over "the

continuous part of the spectrwn. However if we neglect

superposition of other states the sUITilllationin this case is

only over the four states above. may determine the

coefficients ai by Moffitt's method (i.e. from a consideration

of the Wigner-Witmer rules) but it is sometimes simpler to

use the postulate that the expectation value of the snin of

the system must be zero; the reason for this postulate,in

very crude terms, is trl.atany valence electron is being

exchanged continually with one of opposite spin "belonging"•..ao f-kc...\:- Yk.. ""\)('..(''''''.12. :qi",,- A~Sb~.;..t.Q.& ~f1,. rk Qt""'~c..( i~ ~to\8. is u.~

to a peripheral bonded at~This postulate also gives

explicit justification for operating on rows and col-amns of

the determinant above. Since we must con~ider the fi~st three

functions equa~ly the only combination of all four functions

satisfying the postulate is "1.1'61.:

This is a non-stationary state, but if we make a large number

- 7 -

of observations on systems described by this function, we

will find that three quarters of them will give the result

E(3p) and one quarter will give E(lp). The expectation

value of the energy will therefore be

Also, for the p2 configuration, we may choose in a similar

manner a wave function having two random spins i.e.

= tdet.I~1(1)'E2{1) I + tdet.I~1(1)'E2(1)1

j- (2)-idet.I~1 (1) '~2(1) I·

with

Suppose now that neither ~ nor E£ alone gives a good

representation of the valence state of the atom. ~e may

choose as a variation function a linear conwination of the

two sets of determinants, i.e.

- 8 -

11/ a~ + b.:££) ='v

<th 2 1-,2 1Wl _ a + u ::::: • This is II second order hybridisation. The

first point of difference between this and the more familiar

first order hybridisation is that in this the energy expectation

value of the valence state depends on the mixing coefficients viz.

E(a~ + bJ2.I~) _v -

Furthermore, there is only a limited field within which

we may call this process hybridisation. Suppose that we wish

to consider a valence state as a hybrid of s2d and p2~: the

determinants we have to add now differ in two colmnns from

each other and there is no simple addition rule for such» "

determinants. Consequently in the atoms in molecules

approximation to the wave function, the s, p and d atomic- - -orbitals lose their significance. This difficulty applies in

particular to lone pairs in connection with their hybridisation

with bonding orbitals, which is supposed to operate in the

almnonia and water molecules. For if we are not satisfied that

~2pxPyRz gives a good description of the valence state of the

- 9 -

nitrogen atom in aYilinonia,(e.g. because the bond angles are

a 2greater than 90 ), we must postulate that some of the ~ lone

pair has been promoted. But we cannot maintain zero spin

expectation value for the valence state wave function unless

we promote equal amounts of sand s electrons. In other words

InS)we must mix determinants wni~h differrbY more than one column""Ill I'~s t\lef. "'-is ••. ~o •. +e1:h'~S""",\: •••\;i~ j•.• <JlMd. l-hc. ""\r,,,,", .(. 10""-«,, pAir) j •• i"'DW\Si5~4l""" "",,11..

,. "t " _. .•::...J-. " •.....•

~ o•.~t..t.1 o..s.~\>H~", !'>f .f; liM"-I eo•••loi", ••. ~;,:"" bt •••1-0 ••.••• Ci'b.h.ls.or ,".~~ i'1!1t"': IIIJt;ln" \if lone fi!".JIIH.

Hybridisation and Bond Angl~

The use of the orbital picture to account for bond angles

is closely associated with the idea that the individual bonds

in a molecule are largely independent of each other(3), both

in their physical properties and, by analogy, in respect of

the electrons from which they are formed. Considering the

arsenic atom, we illayregard a suita. Ie valence state as

(core + p p p ), i.e. the wave function will behave as-x-y-z

det.lp (1), p (1), p (1)1;-x -y -z

(note that this represents, in the absence of perturbation,

a spherically symInetrical distrilJution of charge). we navv

imagine the approach of three hydrof?:enatoms hI' h2 '--1ndh----' - f_ '-, _, UJ -l..- 3 ,

- 10 -

which pair identically, not i'vith electrons 1, 2, .3, (Jut with

the orbitals p , p , and p! Pauling's original nostulate then-x -y -zleads us to expect that the most stable configuration will

occur when hI' h2 and h3 lie along the x, y and z axes

respectively. In other words arsenic will form three bonds

at right angles (cf. experimental bond angle of 92° in AsH3).

The principal use of first order hybridisation has been

~:E}V\!\l associated with the sinr9le extension of this thought

process to include hybrid orbitals, an occupation which has1proved most useful in rationetsing almost all the observations

in the stereochemical field. feel therefore that it is

illuminating to examine some of the limitations of the treatment.

A severe limitation, and one which has been recognised

for a long time, is that I'vecannot retain simple directional

orbi tals if we mix atomic V'iavefunctions having different

radial dependences. This difficulty is easily avoided by using

Slater functions when the orbitals have the same principal

quantum rrunilier.But if we consider, as we must in many

of orbitals i'l different quantum groups (which have differing

exponential dependences) it is customary to consider only the

angular parts of the functions ~ and ELssurnea uniform

radial dependence for all the functions being mixed. ~~~

- 11 -

As ~"\

~ 1 t' ~.'~f' b ' t" t 3 2d -. t 1?..c...::- examn e ne QLC erence e-cween He wo :;?p_ orbl a. s,

one constructed from Slater functions and thre other from

L-1$ $••0 ••.•" ••••• -F~ •• ~€ Ihydrogen-like orbitals, \OT}l;,.h Hg;Ve ~"Q:'lrI~.L~:::,I~j;;.a:1:::::Q;;~n~_~~cii}

Whilst in the latter case the orbital is still sy~netrical about

the x-axis, it is no longer possible to say that it"points"

~~in a5W~' ··6f~*~ direction like the Slater or1:dtal does.

This does not matter when discussing square complexes because

Pauling's fourlllsquare~Ylybridslistill rem~clinequivalent to each

other and belong to the same sJ'iTIffietrygroup irrespective of

the radial dependences. However, when non-equivalent orbitals

are involved, each hybrid wave function will have different

percentages of §,~ and d character and the directions of the

hybrid orbital maxima will not then be simply related to each

other, i.e. the predicted bond angles will vary with the

distance from the central atom. Hence an LCAO description

of non equivalent directed valencies is only possible if the

radial dependences are irrelevant.

Neglecting radial dependences therefore, how c:an we

determine the form of the n orbitals appropriate to any given

molecule, by this method? Fop example what cOlnbination of R2~

orbitals could be used to describe the curious T-shaped

molecule ClF3? We would have three hybrid orbitals written

- 11-

Legend to Figure I.

Contour maps of W4

:::::1J§ + using

(a) hydrogen-like and (b) Slater 3s, 3p and 3d orl:itals- - -

covering the area -40a /x~,40a • A scale factor (the same- -O~'4 0 '3Z 3Z

in both cases) is ap~olied so that in general -100 ~ II! ~ 100.- ,,- ~-

But for the purposes of pictorial representation of both

positive and negative magnitudes the value W is only printed

if 0~11jr1<L19.-5and a space is left if 11jr1~49.5. Thus the

numbers 00 to 49 represent positive values of 1jrand the

numbers 50 to 99 represeel1t the corresponding negative values

(i.e.17 means nt :::::17, 71 means n1jr:::::- 21 and a space means

<ntl W I ~ 49.5, the sign being obvious from the local environment).

These calculations were performed using the Manchester

University Electronic Computer.

3sp2d hybrid orbital

(hydr"f'en-like)

13 14 15 15 15 16 15 15 14 14 12 11 09 07 05 03 53 55 58 59 59 60 60 60 60 59 59 58 58 57 56 56 55 55 54 53 53 53

15 16 17 17 18 18 18 17 16 15 14 12 1008 06 03 0 52 54 56 5 60 61 61 62 62 62 61 61 60 59 59 58 57 56 56 55 54 54 53 53

17 18 19 20 20 20 20 20 19 18 16 14 11 09 06 03 0 53 56 58 6 62 63 64 64 64 64 63 62 62 61 60 59 58 57 56 55 55 54 53 53

19 21 22 22 23 23 23 22 21'20 18 15 13 09 06 02 51 55 58 606 65 66 66 67 67 66 65 64 63 62 61 60 58 5~ 56 56 55 54 54 53

22 23 24 -25 "26'26'26"25,2422 20 17 14 10 06 0 53 57 60 63 6 68 69 70 70 69 68 67 66 64 63 62 60 59 58 57 56 55 54 54 53/ , '

24 ~6 27 28 29,29 2929 27 2'5. 22 19 15 10 06 1 54 59 63 67 6 71 7J 7J 7J 72 71 69 68 66 64 62 61 60 58 57 56 55 54 54 53/ '" - - -27 29 30 32 33 33 33 32 30 28 25 21 16 11 05 1 57 62 67 71 7 -,677 77 76 75..73 71 69 67 65 63 61 60.5,8 57 56 55 54 54 53\ i . "

30 32 34 35 36 37 37 36 34 31 27''?3 17 11 04;53 59 66 71 15 7 80 81 81 79 77 15 7J 70 68 66 63 62 60 58 57 56 55 54 54 53'

.I. / \

33 35 37 39 40 41 41 40 38 34 30 2~ 18 11 0/ 55 63 70/76 81 8 85 85 84 82 80 77 \,74 71 68 66 6361 60 58 57 56 55 54 53 5336 39 41 43 44 45 45 44 42 38 33 27\ 19 10 (11 58 67 ;(5 82 87 91 90 88 85 82 78 :74 71 68 65 63 61 59 57 56 55 54 54 53 53

39 42 45 47 49 49 49 48 45 41 36 28)19 09 ;51 62 ;/81 89 93 96 94 91 87 82 78;74 70 67 64 61 59' 58 56 55 54 54 53 53 5242 45 48 49 45 38 30120 08 54 67/78 88 96 97 92 87 81 7.6 72 68 64 62 59 58 56 55 54 54 53 53 52 52; I45 49 48 41 31120 06 58 7'1. 85 96 98 92 85 7~ 72 68 64 61 58 57 55 54 53 53 52 52 52 52 51

48 43 32: 19 04 62,78 93 I 96 87 ;/72 66 6158 ~2- 52~_~, 51 51 51 51 51 51

45 33/18 01 67/85 /' 90 7~69 62 5~-00 01 01 01 01 01 ~5D-2£47 34! 17 52 1'1 92 92 ?fi 1,3 )A--Di-~6 07 08 07 07 06 05 04 03 02 02 01 ?1 01 00

49 34[16 55 -i8 ~ 715 /6'9']) 2.Q2D 19 17 15 13 11 09 07 05 04 03 02 02 01 01I i -- "-

34' 15 58 '82 9 -:3040 42 40 36 32 27 22 18 15 12 09 07 06 04 03 02 02 01, I "

3<14, 60;87 43 35 29.23.18 15 12 0907 05 04 03 02 0234:13 62) 89 41 33 26,21 16 13 10 08 06 04 03 02 02

-------.-.--.-'.-.--- .. ---.- )4 '1)' 6)190 4) )4 27122 17 1) 10 08 06 0) 0) 0) 02

34 :13 62! 89 41 33 2821 16 13 10 08 06 04 03 02 02: I /

34 :1460\87 832p 43 35 2~2J 18 15 12 09 07 05 04 03 02 02

34,115\,58 ,~2 7F ~"30_ ~ ~ 40 3~z.-~-V:2 18 15 12 09 07 06 04 03 02 02 0149 34!16 \5518 9p ~5 1720201917151311090705040302020101

, I \ I ~47 34 '17 P2 72 92 , 92 7"" 63 54'-92.....0607 08 07 07 06 05 04 03 02 02 01 01 01 00. I \ \ ) "'- ---___.

45 33 '18 ).1 67, 85 I 90 7<J~9 62 57 5)~ no· 01 01 01 01 01 01tl.{\ '- _.~

48 43 32' 19 O~ 62 ~8 93 96 87 79"72 66 61 58 56 54 53 52 52 51 51 51 51 51 51 51! , "

45 49· 48 41 31 20 O~ 58 ia 85 '96 98 92 85 78' ~2 68 64 61 58 57 55 54 53 53 52 52 52 52 51

42 45 48 49 45 38 30:20 08\54 67\,\8 88 96 97 92 87 81 7\72 6864 62 59 58 56 55 54 54 53 53 52 52

39 42 45 47 49 49 49 48 45 41 36 28119 09 '01 62 72\81 89 93 9F 96 94 91 87 82 78 V4 70 67 64 61 59 58 56 55 54 54 53 53 52

36 39 41 43 44 45 45 44 42 38 33 21 19 10 q1 58 67 75 82 87 9P 91 90 88 85 82 78 V4 71 68 65 63 61 59 57 56 55 54 54 53 53I , '\ I !

33 35 37 39 40 41 41 40 38 34 30 ~5 18 11 O~ 55 63 70 ~6 81 8~ 85 85 84 82 80 77 14 71 68 66 63 61 60 58 57 56 55 54 53 53, "-- ! /303234353637 37 36 34 3127;2317 110 5359667175718 8081817977 7.5 7J 706866636260585756 55 54 54 53

, " /27 29 30 32 33 33 33 32 30 28 2£ 21 16 11 05 51 57 62 67 71 1!~-u, 77 77 7§. -:f5 73 71 69 67 65 63 61 60 58 57 56 55 54 54 53/ ' ---~ ~6 27 28 29 29 29 29 27~5 22 19 15 10 06 1 54 59 63 67 ~9 71 73 73 73 7271 69 68 66 64 62 61 60 58 57 56 55 54 54 5322 23'2~5~6l:...6 ,3.6...P.)'24 22 20 17 14 10 06 0 53 57 60 63 46 68 69 70 70 69 68 67 66 64 63 62 60 59 58 57 56 55 54 54 53

19 21 22 22 23 23 23 22 21 20 18 15 13 09 06 02 51 55 58 60 ~ 65 66 66 67 67 66 65 64 63 62'61 60 5857 56 56 55 54 54 53

17 18 19 20 20 20 20 20 19 18 16 14 11 09 06 03 0 53 56 58 '62 63 64 64 64 64 63 62 62 61 60 59 58 57 56 55 55 54 53 53

15 16 17 17 18 18 18 17 16 15 14 12 10 08 06 03 0 52 54 56 8 '60 61 61 62 62 62 61 61 60 59 59 58 57 56 56 55 54 54 53 53(;'

13 14 15 15 15 16 1515 14 14 12 11 090705 03 01 51 5355' 58 59 '59 60 60 6060 59 59 58 58 5756 56 55 55 54 53 53 53

3Sp2d hybrid orbital

(Slater)

1

52 53 53 54 54 55 56 56 57 5.8 59.. 606061 61 62 62 61 61 60 5 58 5655.53 l1 01 02 04 05 07 08 08 09 09 09 0.9 09 09 09 08

52 5353 54 55 55 56 575859 60 61 62 62 63 63 63.63 62 61 6 595755 53 5 01 03 05 07 08 09 10 11 11 11 11 11 11 10 10

52 53 53 54 55 56 57 58 59 60 61' 62 63 64 64 65 65 64 64 63 6 59 57 55 53 0 02 05 07 09 10 12 13 13 13 14 13 13 13 12 11

52 53 53 54 55 56 57 58 59 60 62 63 64 65 66 66 67 66 65 64 6 '~O 58 55 52 1 04 06 09 11 13 14 15 16 16 16 16 15 15 14 13

52 53 535455 56 57 58 60 61 63 64 65 67 68 68 68 68 67 66 6 61 58 55 52/02 05 09 11 14 16 17 18 191.9 19 1918 17 16 15/ .52 53 53 54 55 56 57 59 60 62 63 65 67 68 69 70 70 70 69 68 6 62 59 55 5,~ 03 07 11 141"7 1921 22 23 23 22 22 21 20 18 17

52 52 53 54 55 56 57 59 60 62 64 66 68 69 71 72 72 72 71 69.6 63 59 54 00 05 10 14 18 21 2J-25"2627272bz5--<~23 21 20

52 52 53 54 54 56 57 58 60 62 64 66 68 70 72 73 7~ 7~ 73 71 6 64 59 5402 08 13 18 22;2628 30.31 31 31 3'0 29 28 2e--2~ 22

51 52 52 53 54 55.56 58 60 62 64 66 69 71 73 7;5/76 761\ 73 6 64 59 57 04 11 17 2);28 31 34 36 37 37 36 35 33 31 29 27 2-§

51 51 52 52 53 54 56 57 59 61 64 66 69 72 74/76 77 77 76 ~4 t 65 58 Sb 07 15 22 -'28 34 38 41 42 43 43 42 40 38 36 33 30 2850 51 51 52 52 54 55 56 58.60 63 66 68 71.74177 78 79 78 7\) 71 64 57- 02 1120;1a 35 41 45 48 49 49 48 45 43 40 37 34 31

~ ~n \ J I .00 u~ 51 51 52 54 55 57 59 61 64 67 71 74,76 79 79 79 7" 11 64 55 05 15 f'5 35 42 49 48 44 41 37 3401 01 010050 51 5253 5.5 57 59 62 65 69 72 7? 78 79 79 7~ 7j1 63 53 09 2)' 32 42 49 45 41 3702 02 01 01 01 00 50 52 53 '55 57 60 63 66 70 73\76 78 78 75 1b '61 01 14 i7.40 - 49 44 40" I: ,

02 02 02 02 02 02 01 01 51 52 54 56 59 63 66 70 73''l6_7fy 74 ~58 04 19134 48 47 42

03 03 03 03 03 03 03 03 02 01 51 53 55 58 62 65 69 72 73 72 ~ 55 09.2~ 42 45

.03 04 04 04 05 05 05 05'04 04 03 01 51 53 56 60 64 67 69 68 ~.52 14 ~2 47

04040505 06 06 06 070706 06 05 04 0251 54 57 60 63 63 ~9 02 1~;J9 49, I

04 05 05 06 07 07 08 08 08 09 08 08 07 06 04 02 505356 58 ~5,06 2f 4505 05 06 06 07 08 08 09 09 10 10 10 10 09 08 07 05 02 ~2 5f1 08 a} 49

05 05 oG 87 87 88 0') 0') 10 10 11 11 11 10 0'}B&·-e6--0'} OJ O~ ~//!\ I

05 05 06 06 07 08 08 09 09 10 10 10 10 09 08 07 05 ~~50 52 ~~ 08 ~7 49

04 05 05 06 07 07 08 08 08 09 08 08 07 06 04.??~ 53 56 58 ; 06 2\4 4504 04 05 05 06 06 06 07 07 06 0.6.. 0..5 04}2-51 54 57 60 63 63 9 02 1~ 39 4903 04 0404 05 05 05 05 04 04 03/0V51 53 56 60 64 67 69 68 52 14 \32 4703 03 03 03 03 03 03 03 02 0~'~1 53 55 58 62 65 69 72 73 72 6 55 09 25 42. 45

/'" \

02 02 02 02 02 02 01 91'51 52 54 56 59 63 66 70 73 .:ro76',74 8 58 04 19.34 48 47 42.. / \

02 02 01 01 010050 52 53 55 57 60 63 66 70 73 16 78 78 75 0 61 01 14 2.7 40 49 44 40/ \ \

01 01 01 00 50 51 52 53 55 57 59 62 65 69 72 ?5 78 79 79 76 1 63 53 09 21 32 42 49 45 41 37.1 \

~50 51 51 52 54 55 57 59 61 6467 71 74/76 7979 79 7~ 1 64 55 05 15 ~ :5 42 49 48 44 41 37 3450 51 51 52 52 54 55 56 58 60 63 66 68 71 74177 78 79 78 7$ 64 5702 11 20~8 35 41 45 48 49 49 48 45 43 40 37 34 31

51 51 52 52 53 54 56 57 59 61 64 66 69 72 74l76 77 77 76 14 65 58 50 07 15 22~8 34 38 41 42 43 43 42 40 38 36 33 30 28\ I "-51 52 52 53 54 55 56 58 60 62 64 66 69 71 73 79...?!']6/75 73 64 59 52 04 11' 17 23 ~ 31 34 36 37 37 36 35 33 31 29 27}552 52 53 54 54 56 57 58 60 62 646668 70 72 73 74 74 73 71' 6 64595402 08 13 18.222-6283031 31 31 3029 28 ~,(4 22. "-52 52 53 54 55 56 57 59 60 62 64 66 68 69 71 72 72 72 71 69 6 63 59 54 00 05 10 14 18 21 24'25..g6 27 27 26 z.s... 2'423 21 20

52 53 53 54 55 56 57 5960 62 63 6567 68 69 70 70 70 69 68 6 '62 59 55 5' 03 07 11 14 17 19 21 2;23' 23 ;; 22 21 20 18 17

52 53 53 54 55 56 57 58 60 61 63 64 65 67 68 68 68 68 67 66 6 61 58 55 52~02 05 09 11 14 16 17 18 19 19 19 19 18 17 16 15

52 53 53 54 55 56 57 58 59 60 62 63 64 65 66 66 67 66 65 64 6 60 58 55 52 1 04 06 09 11 13 14 15 16 16 16 16 15 15 14 13

52 53 53 54 55 56 57 58 59 60 61 62 63 64 64 65 65 64 64 63 59 57 55 53 0 02 05 07 09 10 12 13 13 13 14 13 13 13 12 11

52 53 53 54 55 55 56 57 58 '59 60 61 62 62 63 63 63 63 62 61 59 57 55 53 51,01 03 05 07 08 09 10 11 11 11 11 11 11 10 10

52 53 53 54 54 55 56 56 57 58 59 60 60 61 61 62 62 61 61 60 5 58 56 55 53 51\01 02 04 05 07 08 08 09 09 09 09 09 09 09 08

- 13 -

= alk p + ank p + a31 d,-x c. -y -,{- where k

d is maximal 1.n the X7:l plane. Applying Pav.ling J s criterion

we then have that dWl/~~' ~W2/d& and ~w3/a&must be zero

for e = 870, 00 and -870 respectively, giving rise to three

equations for the co:efficients a.,. Also since both the component. In: -

orbi tals rare normaiiSed 1:.nd tl:~~,hY~:~~~, there are another

five independent equations involving the coefficients giving

altogether eight equations for nine coefficients. However

the exclusion principle is not obeyed if the tk are not(L)

orthogonal to each other ' , so that there are three more equations

i.e. in all we have eleven equations for n1.ne unknowns! The

system is over specified and there is no physically acceptable

solution for the coefficients; all one can do is relax some

of the restrictions to reduce the number of equations to nine,

e.g. two of the orthogonality relations when a solution can

be obtained which does not obey the exclusion principle.

In the general(i.e. three dimensional case) it is only

for n=6 that an acceptable unique solution exists; if n ~5

the situation described aGove is encountered, and for n ~7

there are not enough equations from which to det4rmine the

coefficients.

-·l,t -

We conclude therefore, that~he hybridised orbital

picture does not account satisfactorily for the occurrence

of directed valence. As we see the situation,it is generally

believed that in some way the bond angles in a cOTI~ound IvTHnare essentially determined by the electronic properties of

the central atom. These "electronic properties" are usually

taken to mean orbitals; but we must be careful to what extent

we base physical theories on these constructs: for an atom

in a field free vacuum. a directional orbital can have no

significance as it is not invariant to a rotation of

coordinates in contrast to the Hamiflonian of the system.

It is only in the presence of a perturbation' (i.e. the other

atoms) that the concept of orbitals is useful. But it is

curious that the approach of a hydrogen atom to a sulphur

atom causes it to develop another orbital suitable for

bonding at right angles to the line of approach, whereas

a boron atom develops two in the same plane at trigonal

angles; especially so when perturbation theory tells us that

a pertur1Jation along the z axis affects only th.ose orlJitals

which are symmetric about that axis(5). The most puzzling

feature about contemporary thought is the unquestioning Q(U.p~A""c:e

of s, p, d etc. orbitals which appear from the solution of the- - -

wave equation for the hydrogen atom in spherical polar

l~ -- ./

coordinates, v/hen the problem can be just as eflsily solved in

parabolic coordinates(6) (such solutions have been used in

discussing the Stark effect and would appear formally to be

more appropriate in discussing diatomic molecules than the

more familiar ones). One often sees in the literature the

phrase lIapure p bond". But what virtue has the p wave function

except that it is an eigenstate of the angular momentum etc.?

In any case we are not particularly interested in angularP+O~&I~~

momentum, for when we hybridise orbitals this ~~ becomes

indeterminate. The great advantage of the functions is that

they can be combined easily to form three equivalent orthogonal

p orbitals which are real functions but it is unthinkable that

thii mathematical convenience should confer any ~X~~~

physical reality to lIalIUre p bondtl in our discussion of

molecular binding.

How then can we account for the direction of valence

and the contrasting behaviour of elements in the left and

right hand sides of the periodic table? The answer is indeed

a very simple one, which is independent of any concept of

orbitals. First we consider the s~fietrical approach of two

protons Ip1 to a hydrogen like atom M ( i.e. with the M-P

distances equal): the energy of repulsion between the nuclei

will be a minimum when they are in a straight line P-l\1-P;

- 16 -

but the energy of attraction between the electron and the

three nuclei is a maximuIIlwhen the two P nuc .....lei are superimposed,..."

i. e. [2P] The observed bond angle will depend on the

absolute magnitudes of these two energy ~erms/and the way in

which they vary/with the P-M-P angle. We may divide the

comnounds MH into two classes, (1) where the nuclearnrepulsion term dominates and (2) where the net electronic

attraction term dominates. If we consider the approach of

two protons to a Be ion to form BeH2, the nuclear repulsion

term dominates and a linear molecule is formed f,similarlvu

BH3 will be trigonal and CH4 tetrahedral because these

arrangements minimise the nuclear repulsion energy for a

.. t 1 .1-' )glven ln ernuc ear separaulon . However if the two protons

approach 0-- having ten electrons confined to :J. rather smaller

volume than the six electrons in Be-- (the oxygen atom is

smaller than the Be atom and we have assUIned that this is

also true for the corresponding negative ions), they encounter

a TIlliCngreater density of charge which makes the net attraction

term dominate; the effect of this is to reduce the H-O-H

angle from 1800 to 1040, where the attractive force tending

to superimpose the two nuclei balances the nuclear repulsion.

This approach to the prOblem of directed valence means that

for case (I) molecules (M belonging to the left side of the

neriodic table) we have a COIT@on~

- 17 -

explanation of the bond angles which occur, but for case (2)

molecules ( • ec-l.e. I'll on the right hand side of the periodic

table) we have to consider each one separately. There are

however some generalisations about case (2) which can be made,

and which are supported by experimental observation. For

examp~e, the greater the electron density around the central

atom, the smaller will be the bond angle(cf. the successive

reduction in bond angle along the series NH3, PH3, AsH3 and

SbH3); or again an atom NT in a low valence state may belong to

case (2) but in a higher valence state it may pass over into

case (1) because of the increasing number of nuclear repulsion

terms (e.g. H2S is angular but in SF6 the fluorine nuclei take

up a s~nmetrical arrangement of minimUTIlnuclear repulsion);:!l~a.,.

also, ~~~ is case (1) whereas lower down the table TeCl4

is a case (2) molecule in accordance with the nrevious

generalisatio~. Thus although the reason for the phenomenon

of directed valence is conceptually much simpler than hitherto

realised, the problem of calculating case (2) bond angles isJ..o~",~.4

much greater; ~ we believe that useful qualitative

results ~ lJe o1::,taine~by- treating the central atom as a

semi-classical field. The above treatment, of course, ignores

the kinetic energy of the electrons which does not appear to

be important, at least in determining general trends (c.f· o.lso -Hi- 7)

- 18 -

Hybridisation and Electric Dipole Iiloments.

In recent years the discussion of dipole moments in terms

of hybridisation has become cOEMon, although no clear

differentiation seems to have been made between the effect of

first and second order hYbridisation~. The irrelevance of

first order hybridisation to molecular dipoles has already

(g) "<jl.rioti"'''''l-ioV'lbeen discussed qualitatively • First orderAdoes not alter

the total wave function of an atom and cannot therefore confer

upon it any polar character. We do not deny however that it

is sometimes convenient to discuss the dipole moment of a

particular bond and in this respect the hybrid nature of the

bonding orbitals is important.

It is only second order hybridisation which changes the

electron distribution in an atom and we must enquire therefore

to what extent the occurrence of dipole moments can be

described in su.ch terms. Consider the very simf)le case of thelil'\.•••.••.•

~ atom where we wish to represent the valence state as a') ')

hybrid between ls~2s and 1~~2p. Since these two representations

only differ in the state of one electron we may Tetain the

orbital picture and wTite the valence state ~~ve function as

a linear combination of two single terms, i.e.

- 19 -

Because sand p wave functions have different parities this

valence state has an "atomic dipole", in say, the +x direction

given by

x = Jw XW dT = a2 Jsxsd~ + 2ab Jsxpdr + b2 Jpxpdrv 'v - - -

= 2ab Jsxndr- ;;;

since the individual ~ and :p orbitals have no dipoles themselves.

But we could equally well take

or even

ibp (1);:.x

the first having a dipole in the opposite direction to Wv and

the last two having no dipole at all, but a+l of which, along

with an infinite nUD~ber of others, have the same expectation

value for the energy as the conventional Wv. In other words

by altering the phase of the wave function we can alter the

physical state which it represents, which is not permissible;

we must average over all possible phases of the wave functions

- 20 -

q

in which case the interference between them vanishesC').

have therefore to consider the question as to what

extent second order hybridisationhas any physical significance.

We have not included the energy dependence of the ·two

orbitals which would cause the electron distribution to

oscillate with a fEequency corresponding to the difference

in energy of the two states. All we are trying to do is to

construct a power series in r, e and ¢ which represents the

electron distribution in the atom as it is in the molecule,

and the spectroscopic states of the free atom are a convenient

set of functions for this purpose/. In which case it is clear

that we must not attach too much physical significanc:e to

the above concepts which are the offspring of the type of

functions we choose as our starting point, in much the same

way that llexchange integrals" etc. arise from the 1C1\.0 method;

these are not invariant under transformations of the initial

functions chosen and do not appear at all in the James-

Coolidge treatment of diatomic molecules. The concepts

arising out of the treatments discussed in this paper are

of a similar nature and in consequence should not be taken too

h~ I"' C'""

- 2~ -

1. L. Pauling, J. Amer. Chem.~ Soc., 1931, 53, 1367.-, ,-2 w 'en f-~·tt h 1=1 P "011-' 10h4 17 '73• 'I. 1\"0 Il , ~'1.nn. l.,ep. rag" .J. __ ,\TSlCS, j.l, _,.L .3. C.J:\..Coulson, Valence, Oxford. University Press, 1952.

4. J.E. Lennard-Jones and J.A. Pople, Disc. Faraday Soc. 1951,10, 9,

5. L.I. Schiff, Quantum Mechanics, McGraw Hill Co., N.Y., 19~9.

6. E.U. Condon and P.M. Morse, QuantUli1 chanics, Graw Hill Co. N."/.}

1929.

7. M.A. Cook, J. Chem. Phys., 1£, 62, 1946.

8. H.O. Pritchard, Disc. Faraday Soc., 19,276, JCj5"S".

9. 1v1. Born, Continuity, Determinism and Reality, Det. Kgl. Danske

Videnskab Selskab, th-fys-Medd, 1955, ]Q, No.2, 25.

''Jr, _ A]:;':D,,~~l~ g,:j;:!g g g Q';P"''''T,\*.0",,"C'<~._Jl1IC\1. c@GjJJ.@~ .<:lYl..=l,.{:\u",,,,+,,,~·""'-'ft'tr'~-~~~ :t: ••• III ~.f III ~~1.'ll'0·'''r~>~~;:t~~---· .•... ",.,;j';;J"~:t..t;:':,,,,~