Rauk Orbital Interaction Theory of Organic Chemistry

	LCAO SOLUTION OF FOCK EQUATIONS	229
The quantity in large parentheses is the Fock operator, F …1†,
F …1† ˆ h…1† ‡	Ne	…A:41†
	‰Jb…1† ÿ Kb…1†Š
	bˆ1
	X

Therefore, the condition that the orbitals yield a stationary point (hopefully a minimum) on the energy hypersurface with respect to variations is that the orbitals are eigenfunctions of the Fock operator, with associated orbital energy, e,

F…1†fa…1† ˆ eafa…1† …A:42†

In summary, to obtain a many-electron wave function of the single determinantal form [equation (A.12)] which will give the lowest electronic energy [equation (A.14) or (A.27)], one must use one-electron wave functions (orbitals) which are eigenfunctions of the oneelectron Fock operator according to equation (A.42). There are many, possibly an in®- nite number of, solutions to equation (A.42). We need the lowest Ne of them, one for each electron, for equation (A.12) [or (A.27)]. When the Ne MOs of lowest energy satisfy equation (A.42), then E0 Ð EHF [equation (A.27)] and F0 Ð FHF [equation (A.12)].

LCAO SOLUTION OF FOCK EQUATIONS

We must now bite the bullet and specify what form the MOs must have. We expand the MOs as a linear combination of a number of linearly independent functions, the basis set:

Xn

fa…1† ˆ	wi…1†cia	f ˆ wc …in matrix form†	…A:43†
	iˆ1

Such an expansion can always be made without approximation if the set of functions is mathematically complete. We must necessarily use a ®nite (and therefore incomplete) set. We will discuss the characteristics of the basis set below. For now let us take the wi as known and proceed to determining the expansion coe½cients cia. Substitution of equation (A.43) into equation (A.42) yields

X	X
n	n
F …1† wi…1†cia ˆ ea	wi…1†cia	…A:44†
iˆ1	iˆ1

Multiplication on the left by wj and integration over the range of the coordinates of the electron give

X		… wj …1†F…1†wi…1† dt1cia ˆ ea		X	… wj …1†wi…1† dt1cia
i	n1			i n1		…A:45†
	ˆ			ˆ
or		X	X
		n	n			…A:46†
		Fjicia ˆ	iˆ1	Sjiciaea
		iˆ1

230	DERIVATION OF HARTREE±FOCK THEORY
Equation (A.46) may be cast as a matrix equation
	Fc ˆ Sce	…A:47†
The overlap matrix S is de®ned as
	…
	Sij ˆ wi…1†wj …1† dt1	…A:48†

The basis functions are normalized so that Sii ˆ 1, but are not orthogonal, that is, Sij 0 0 in general.

The Fock matrix F is given as

…

Fij ˆ wi…1†F…1†wj …1† dt1

ˆ wi…1†"h…1† ‡		Ne	‰Jb…1† ÿ Kb…1†Š#wj …1† dt1
	…	b 1
		X
ˆ		ˆ		‡ bNe1	… wi…1†Jb…1†wj …1† dt1 ÿ … wi…1†Kb…1†wj …1† dt1
	… wi…1†h…1†wj …1† dt1
				X
ˆ	… wi…1†h…1†wj …1† dt1			ˆ	…… wi…1†fb…2† r12 fb…2†wj …1† dt2 dt1
				‡ b 1
				X
				Ne			e2
				ˆ	ÿ	…… wi…1†fb…2† re12 wj …2†fb…1† dt2 dt1					…A:49†
						2

To construct the Fock matrix, one must already know the molecular orbitals (!) since the electron repulsion integrals require them. For this reason, the Fock equation (A.47) must be solved iteratively. One makes an initial guess at the molecular orbitals and uses this guess to construct an approximate Fock matrix. Solution of the Fock equations will produce a set of MOs from which a better Fock matrix can be constructed. After repeating this operation a number of times, if everything goes well, a point will be reached where the MOs obtained from solution of the Fock equations are the same as were obtained from the previous cycle and used to make up the Fock matrix. When this point is reached, one is said to have reached self-consistency or to have reached a self-consistent ®eld (SCF ). In practice, solution of the Fock equations proceeds as follows. First transform the basis set fwg into an orthonormal set flg by means of a unitary transformation (a rotation in n dimensions),

X			X X			…		†
n			n	n
lj ˆ i	1 wiuij	Sijl ˆ … li…1†lj …1† dt1 ˆ k 1 l 1 ukiSklw ulj ˆ dij					A:50
ˆ			ˆ	ˆ
		Sl ˆ uTSwu ˆ I
The inverse transformation is given by				X		…		†
X		X X
n		n	n	n	1…uÿ1†ikT ukjÿ1
wj ˆ i 1 liuijÿ1 Sijw ˆ		… wi…1†wj …1† dt1 ˆ k 1 l 1 ukiÿ1Skll uljÿ1 ˆ k					A:51
ˆ		ˆ	ˆ	ˆ

Sw ˆ …uÿ1†Tuÿ1

INTEGRALS 231

Substitution of the reverse transformation into the de®nition for the Fock matrix yields

X X		… wk…1†Fwl …1† dt1 uljÿ1	…		†
n	n
Fijw ˆ … wi…1†F …1†wj …1† dt1 ˆ k 1 l 1 …uÿ1†ikT				A:52
ˆ	ˆ

Fw ˆ …uÿ1†TFluÿ1

Substitution of equations (A.50) and (A.51) into equation (A.46) and multiplication on the left by cT yield

Fwc ˆ Swce	…A:47†
…uÿ1†TFluÿ1c ˆ …uÿ1†Tuÿ1ce	…A:53†
cT…uÿ1†TFluÿ1c ˆ cT…uÿ1†Tuÿ1ce	…A:54†
VTFlV ˆ VTVe ˆ e V ˆ uÿ1c	…A:55†

Thus the Fock matrix in the l basis is diagonalized by standard methods to yield the MO energies e and the matrix V from which the coe½cient matrix c may be obtained by c ˆ uV. There are several ways in which the matrix u and its inverse may be determined. The most commonly used is the symmetric othogonalization due to LoÈwdin, which involves diagonalization of the overlap matrix. We will not discuss this further.

INTEGRALS

Solution of the Fock equations requires integrals involving the basis functions, either in pairs or four at a time. Some of these we have already seen. The simplest are the overlap integrals, stored in the form of the overlap matrix S, whose elements are given by equation (A.48) as

…

Sij ˆ wi…1†wj …1† dt1

The Fock integrals ®rst encountered in equation (A.45) are constructed from kinetic energy integrals, nuclear-electron attraction integrals, and two-electron repulsion integrals, as follows, continuing from equation (A.49):

ij ˆ …

i… †" 2m

… †# j … † 1

‡ …

i… †"I 1

ÿr1I

# j … † 1

‡ k 1 l 1

b 1

kb lb

X

ZI e2

X XX

F

w 1

ÿq2

`2 1 w 1 dt

w 1

NN

w 1 dt

n

n

Ne

c c

ˆ

ˆ

ˆ

ˆ

…A:56†

…… wi…1†wk…2† r12 wl …2†wj …1† dt2 dt1 ÿ …… wi

…1†wk…2† r12 wj …2†wl …1† dt2 dt1

e2

e2

X X

ˆ Tij ‡ Vijne ‡

n

n

kˆ1

Pkl Gijkl

…A:57†

lˆ1

The kinetic energy integrals are collected as the matrix T, whose elements are de®ned by

232 DERIVATION OF HARTREE±FOCK THEORY

	ij ˆ	2m …			i… †	… †	j …	†	1	…	†
T		ÿq2		w	1 `2	1 w	1		dt		A:58

The nuclear±electron attraction integrals are collected as the matrix Vne, whose elements are de®ned by

	X
	NN			1
Vijne ˆ ÿ	I	ˆ	1 ZI e2 … wi…1†	r1I	wj …1† dt1	…A:59†

The supermatrix G, which contains the two-electron repulsion integrals, has elements de®ned by

Gijkl ˆ	…… wi…1†wk…2† r12 wl …2†wj …1† dt2 dt1		ÿ	…… wi…1†wk…2† r12 wj …2†wl …1† dt2 dt1		…A:60†
		e2			e2

In equation (A.57) we also introduced a useful matrix, the elements are de®ned by

XNe

Pij ˆ ciacja aˆ1

density matrix P, whose

…A:61†

where the sum runs over all of the occupied MOs. One of the limiting factors in ab initio MO calculations is the computation and possibly storage and reading of the twoelectron integrals. Their number is approximately proportional to n4, where n is the size of the basis set. It is highly desirable to keep n as small as possible! Much care must be taken in the choice of basis set. The choice of basis set has been called the original sin of computational quantum chemistry.

THE BASIS SET (STO-3G, 6-31G*, AND ALL THAT)

The requirement that the basis functions should describe as closely as possible the correct distribution of electrons in the vicinity of nuclei is easily satis®ed by choosing hydrogen-like atom wave functions, h, the solutions to the SchroÈdinger equation for oneelectron atoms for which exact solutions are available:

h	1	N h f	r	; y; f	eÿzj r1I	…	A:62
	j … † ˆ	j	… 1I	†			†

Unfortunately, the exponential radial dependence of the hydrogenic functions makes the evaluation of the necessary integrals exceedingly di½cult and time consuming for general computation, and so another set of functions is now universally adopted. These are Cartesian Gaussian functions centered on nuclei. Thus, gj …1† is a function centered on atom I:

gj …1† ˆ Nj …x1 ÿ XI †nx …y1 ÿ YI †ny …z1 ÿ ZI †nz eÿaj r12I

…A:63†

The superscripts, nx, ny, and nz, are simple positive integers or zero. Their values determine whether the function is s-type …nx ˆ ny ˆ nz ˆ 0†, p-type (nx ‡ ny ‡ nz ˆ 1 in three

		INTERPRETATION OF SOLUTIONS OF HF EQUATIONS		233
TABLE A.1. The STO-3G Basis Set Corresponding to an STO Exponent of Unity
a1s	d1s	a2sp	d2s	d2p
0.109818	0.444635	0.0751386	0.700115	0.391957
0.405771	0.535328	0.231031	0.399513	0.607684
2.22766	0.154329	0.994203	ÿ0.999672	0.155916

ways), d-type (nx ‡ ny ‡ nz ˆ 2 in six ways), and so on. Speci®cally, for a nucleus situated at the origin of coordinates,

	8a3	1=4					2
g1s…1† ˆ					eÿar1
	p3
	128a		5		1=4		2
g2px …1† ˆ						x1eÿar1
	p3
	2048a7				1=4		2
g3dxy …1† ˆ							x1 y1eÿar1
	p3

The correct limiting radial behavior of the hydrogen-like atom orbital is as a simple exponential, as in (A.62). Orbitals based on this radial dependence are called Slater-type orbitals (STOs). Gaussian functions are rounded at the nucleus and decrease faster than desirable (Figure 2.2b). Therefore, the actual basis functions are constructed by taking ®xed linear combinations of the primitive Gaussian functions in such a way as to mimic exponential behavior, that is, resemble atomic orbitals. Thus

Xng

wi…1† ˆ	gj …1†dji	…A:64†
	jˆ1

where all of the primitive Gaussian functions are of the same type and the coe½cients dji are chosen in such a way that w resembles h, that is, has approximate exponential radial dependence (Figure 2.2c).

The STO-nG basis sets are made up this way. Table A.1 gives the STO-3G expansions of STOs of 1s, 2s, and 2p type, with exponents of unity. To obtain other STOs with other exponents x, one needs only to multiply the exponents of the primitive Gaussians given in Table A.1 by the square of x.

A similar philosophy of contraction is applied to the ``split-valence'' basis sets. More ¯exibility in the basis set is accomplished by systematic addition of polarization functions to the split-valence basis set, usually the 6-31G basis. These are designated 6-31G(d) and 6-31G(d,p). These are described more fully in Chapter 2 and are illustrated in Figure 2.3

INTERPRETATION OF SOLUTIONS OF HF EQUATIONS

Orbital Energies and Total Electronic Energy

Solution of the HF equations yields MOs and their associated energies. The energy of fa is

234 DERIVATION OF HARTREE±FOCK THEORY

X
ea ˆ … fa…1†F…1†fa…1† dt1 ˆ ha ‡ bNe1 …Jab ÿ Kab†	…A:65†
ˆ

where the integrals ha, Jab, and Kab were de®ned in equations (A.19), (A.25), and (A.26), respectively. The orbital energy is the kinetic energy of a single electron with the distribution speci®ed by the MO, its attraction to all of the nuclei, and its repulsion in an averaged way with all of the other electrons in the molecule. The total electronic energy in terms of the same integrals was de®ned in equation (A.27) as

X		XX
Ne	1 Ne Ne
EHF ˆ aˆ1 ha ‡	2	aˆ1 bˆ1 …Jab ÿ Kab†
It is clear that		XX
X
Ne	1 Ne Ne
EHF ˆ aˆ1 ea ÿ	2	aˆ1 bˆ1 …Jab ÿ Kab†	…A:66†

The total electronic energy is not simply the sum of the orbital energies, which by themselves would overcount the electron±electron repulsion.

RESTRICTED HARTREE±FOCK THEORY

The version of HF theory we have been studying is called unrestricted Hartree±Fock (UHF) theory. It is appropriate to all molecules, regardless of the number of electrons and the distribution of electron spins (which specify the electronic state of the molecule). The spin must be taken into account when the exchange integrals are being evaluated since if the two spin orbitals involved in this integral did not have the same spin function, a or b, the integral value is zero by virtue of the orthonormality of the electron spin functions

… a…1†a…1† ds1 ˆ … b…1†b…1† ds1 ˆ 1

… a…1†b…1† ds1 ˆ … b…1†a…1† ds1 ˆ 0 …A:67†

As it happens, if a molecule has the same number of electrons with spin up …a† as with spin down …b†, the solution of the HF equations in the vicinity of the equilibrium geometry and for the ground electronic state yields the result that the spatial part of the MOs describing a and b electrons are equal in pairs. In other words, for the vast majority of molecules (F2 is an exception), the HF determinantal wave function may be written as

FRHF…1; 2; 3; . . . ; Ne† ˆ …Ne!†ÿ1=2jf10 …1†a…1†f10 …2†b…2†f20 …3†a…3† fM0 …Ne†b…Ne†j

…A:68†

which yields the familiar picture of MOs ``occupied'' by two electrons of opposite spins. Here M is the number of doubly occupied MOs, that is, M ˆ 12 Ne. If one reformulates the HF equations and total energy expression for a wave function which must have the form of equation (A.68), then one is doing restricted HF (RHF) theory. There are con-

RESTRICTED HARTREE±FOCK THEORY

235

siderable computational advantages to RHF theory, so, unless one has some reason to suspect that the RHF solution is not the lowest energy solution, RHF is the obvious starting point. The RHF electronic energy is

	X	XX
ERHF ˆ 2	M	M M	…A:69†
	ea ÿ	…2Jab ÿ Kab†
	aˆ1	aˆ1 bˆ1
and the MO energy is given by		X
ea ˆ … fa0 …1†F…1†fa0 …1† dt1 ˆ ha ‡ bM1 …2Jab ÿ Kab†			…A:70†
		ˆ
Alternative formulations of the total RHF electronic energy are
	X	XX
ERHF ˆ 2	M	M M	…A:71†
	ha ‡	…2Jab ÿ Kab†
	aˆ1	aˆ1 bˆ1
and	X
	M		…A:72†
ERHF ˆ		…ha ‡ ea†

aˆ1

Notice that the energy of the HF determinantal wave function, equation (A.68), and for that matter for any single determinantal wave function, can be written by inspection: Each spatial orbital contributes ha or 2ha according to its occupancy, and each orbital contributes 2J ÿ K in its interaction with every other molecular orbital. Thus, the energy of the determinant for the molecular ion, M‡, obtained by removing an electron from orbital o of the RHF determinant, is given as

FRHFM‡ …1; . . . ; Ne ÿ 1†
	ˆ ‰…Ne ÿ 1†!Šÿ1=2jf10 …1†a…1†f10 …1†b…1†
	fo0 …o†a…o†fo0‡1…o ‡ 1†a…o ‡ 1† fM0 …Ne ÿ 1†b…Ne ÿ 1†j				…A:73†
is given by	X	X X		X
ERHFM‡	M	M M		M
	ˆ 2 ha ‡ ho ‡		…2Jab ÿ Kab† ‡	…2Jbo ÿ Kbo†	…A:74†
	a0o	a0o b0o		b0o
The energy of the molecule itself, equation (A.71), could have been written as
	X	X X	X
ERHFM ˆ 2	M	M M		M
	ha ‡ 2ho ‡	…2Jab ÿ Kab† ‡ 2		…2Jbo ÿ Kbo† ‡ Joo	…A:75†
	a0o	a0o b0o	b0o

Then the energy di¨erence becomes

236 DERIVATION OF HARTREE±FOCK THEORY

	X
ERHFM‡ ÿ ERHFM ˆ ÿho ÿ	M
	…2Jbo ÿ Kbo† ÿ Joo
	b0o
	X
	M
ˆ ÿho ÿ …2Jbo ÿ Kbo†
	bˆ1
ˆ ÿeo	…A:76†

Thus the ionization potential corresponding to removal of the electron from occupied MO o is just the negative of that MO's energy. This observation is known as Koopmans' theorem. One can similarly show that the energy of the lowest unoccupied MO is an estimate of the electron a½nity of the molecule. In fact, ionization potentials estimated by Koopmans' theorem are fairly accurate, but the electron a½nities calculated this way are much less so.

MULLIKEN POPULATION ANALYSIS

The Mulliken population analysis is a simple way of gaining some useful information about the distribution of the electrons in the molecule. Let us assume again a UHF wave function:

X	… fa…1†fa…1† dt1
Ne ˆ aNe1
ˆ			… wi…1†wj …1† dt1
XX X
ˆ i n1 j n1		aNe1 ciacja
ˆ	ˆ	ˆ

Xn Xn

ˆPij Sij

iˆ1 jˆ1

X			X
n		where Pi ˆ	n	…A:77†
ˆ	Pi		Pij Sij
iˆ1			jˆ1
X			X
NN		where PI ˆ		…A:78†
ˆ	PI		I Pi
Iˆ1			i

In equation (A.77) is de®ned the atomic orbital population Pi. Summing all of the Pi that belong to the same atom, I, yields the atomic population PI [equation (A.78)]. The net charge qI on atom I is just the di¨erence between the nuclear charge ZI and the atomic population,

qI ˆ ZI ÿ PI

…A:79†

DIPOLE MOMENTS

The quantum mechanical dipole moment operator is equivalent to the classical dipole moment due to a collection of point charges,

CONFIGURATION ENERGIES

237

X	X
Ne	NN	…A:80†
m^ ˆ ÿ eri ‡ ZI eRI
iˆ1	Iˆ1

Notice that in equation (A.80) the dipole moment operator and the distances ri and RI are vectors which are usually expressed in Cartesian coordinates. The molecular dipole moment within the BO approximation is evaluated as an expectation value [recall equation (A.8)],

m ˆ … FHF…1; 2; . . . ; Ne†m^FHF…1; 2; . . . ; Ne† dt
			X				X
			Ne				NN
ˆ … FHF…1; 2; . . . ; Ne† i				1 eriFHF…1; 2; . . . ; Ne† dt ‡ I				ˆ	1 ZI eRI
X			ˆ		X
	… fa…1†er1fa…1† dt1 ‡
N					NN
ˆ a e1					I	ˆ	1 ZI eRI
ˆ							X
XX
n	n						NN
ˆ i 1 j		1 Pij	… wi…1†er1wj …1† dt1 ‡ I 1 ZI eRI						…A:81†
ˆ	ˆ						ˆ

The derivation of the second line of equation (A.81) follows the same reasoning as was used to obtain the one-electron part of the electronic energy [equation (A.21)], since both m and h are sums of single-particle operators. The dipole moment integrals over basis functions in the last line of equation (A.81) are easily evaluated. Within the HF approximation, dipole moments may be calculated to about 10% accuracy provided a large basis set is used.

TOTAL ENERGIES

The total energy is the sum of the total electronic energy and the nuclear±nuclear repulsion,

NXN ÿ1 XNN ZI ZJ e2

E ˆ EHF ‡ …A:82†

Iˆ1 JˆI‡1 RIJ

Since the second term is constant for a given geometry, the total energy depends on the choice of basis set through the HF energy. This dependence is illustrated in Table 2.1.

CONFIGURATION ENERGIES

Allen has suggested that the familiar two-dimensional periodic table of the elements has a missing third dimension, with units of energy [108, 317, 318]. In part, he reasons that the elements of the periodic table are grouped according to valence electron con®g- urations by quantum numbers n and l, which indicate orbital size and shape but whose primary purpose is to specify energy. It is proposed that the missing third dimension is

238 DERIVATION OF HARTREE±FOCK THEORY

the con®guration energy (CE) (also previously called spectroscopic electronegativity [318]), the average one-electron valence shell energy of a ground state free atom, which may be de®ned as follows:

CE	ˆ	aes ‡ bep			…	A:83
		a	‡	b		†

where a and b are the occupancies of the valence shell s and p orbitals, respectively, and es and ep are the multiplet-averaged s and p shell ionization potentials. The latter may be measured spectroscopically or identi®ed by Koopmans' theorem with the atomic orbital energies. For the d-block transition elements, a parallel de®nition applies, namely,

CE	ˆ	aes ‡ bed			…	A:84
		a	‡	b		†

although the occupancy of the d shell may be di½cult to assign. Values of CE closely parallel the established electronegativity scales of Pauling [319] and Allred and Rochow [320]. A comparison of the three electronegativity scales for selected main group elements is presented in Table A.2 [321].

TABLE A.2. Comparison of Con®guration Energy with Electronegativity Scales of Pauling (wP) and Allred and Rochow (wA&R)a

	H
CE	2.300
wP	2.20
wA&R	2.20
	Li	Be	B	C	N	O	F	Ne
CE	0.912	1.576	2.051	2.544	3.066	3.610	4.193	4.787
wP	0.98	1.57	2.04	2.55	3.04	3.44	3.98
wA&R	0.97	1.47	2.01	2.50	3.07	3.50	4.10
	Na	Mg	Al	Si	P	S	Cl	Ar
CE	0.869	1.293	1.613	1.916	2.253	2.589	2.869	3.242
wP	0.93	1.31	1.61	1.90	2.19	2.58	3.16
wA&R	1.01	1.23	1.47	1.74	2.06	2.44	2.83
	K	Ca	Ga	Ge	As	Se	Br	Kr
CE	0.734	1.034	1.756	1.994	2.211	2.424	2.685	2.966
wP	0.82	1.00	1.81	2.01	2.18	2.55	2.96
wA&R	0.91	1.04	1.82	2.02	2.20	2.48	2.74
	Rb	Sr	In	Sn	Sb	Te	I	Xe
CE	0.706	0.963	1.656	1.824	1.984	2.158	2.359	2.582
wP	0.82	0.95	1.78	1.96	2.05	2.10	2.66
wA&R	0.89	0.99	1.49	1.72	1.82	2.01	2.21

a Ref. 318.