Rogers Computational Chemistry Using the PC

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We do not know either side of Eq. (6-33), but we do know that E is to be minimized with respect to some minimization parameters. The only arbitrary parameters we have are the a1 and a2, which enter into the LCAO. Thus our normal equations are

	qE	¼ 0	ð6-34aÞ
	qa1
and
	qE	¼ 0	ð6-34bÞ
	qa2
These minimizations lead to
	a1a þ a2b ¼ Eða1S11 þ a2S12Þ		ð6-35aÞ
and
	a1b þ a2a ¼ Eða1S12 þ a2S22Þ		ð6-35bÞ
or
	a1ða ES11Þ þ a2ðb ES12Þ ¼ 0		ð6-36aÞ
and
	a1ðb ES12Þ þ a2ða ES22Þ ¼ 0		ð6-36bÞ

A further simplification is made. The wave functions p1 and p2, which are orthogonal and normalized in the hydrogen atom, are assumed to retain their orthonormality in the molecule. Orthonormality requires that

		ð		ð
S11	¼ S22 ¼	p1p1 dt ¼		p2p2 dt ¼ 1	ð6-37aÞ
and		ð		ð
S12	¼ S21 ¼	p1p2 dt ¼		p2p1 dt ¼ 0	ð6-37bÞ
This yields
ða EÞa1 þ b a2			¼ 0		ð6-38aÞ
and
b a1 þ ða EÞa2			¼ 0		ð6-38bÞ

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as the normal equations having the solution set fa1; a2g. In this context, the normal equations are also called secular equations. The exchange integral b is sometimes called the resonance integral.

Homogeneous Simultaneous Equations

What we formerly called the nonhomogeneous vector (Chapter 2) is zero in the pair of simultaneous normal equations Eq. set (6-38). When this vector vanishes, the pair is homogeneous. Let us try to construct a simple set of linearly independent homogeneous simultaneous equations.

x þ y ¼ 0	ð6-39aÞ
x þ 2y ¼ 0	ð6-39bÞ

These equations cannot be true for any solution set other than {0, 0}. The determinant of the coefficients is not zero

	2	1		¼ 2 1 ¼ 1	ð6-39cÞ
	1	1

Any linearly independent set of simultaneous homogeneous equations we can construct has only the zero vector as its solution set. This is not acceptable, for it means that the wave function vanishes, which is contrary to hypothesis (the electron has to be somewhere). We are driven to the conclusion that the normal equations (6-38) must be linearly dependent.

Linearly dependent sets of homogeneous simultaneous equations, for example,

x þ 2y ¼ 0	ð6-40aÞ
2x þ 4y ¼ 0	ð6-40bÞ

are true for any solution set you care to try. They have infinitely many solution sets. The determinant of the coefficients of linearly dependent homogeneous simultaneous equations is zero. For example,

	1		2	¼ 4 4 ¼ 0		ð6-40cÞ
	2		4
which is to say					that the matrix of coefficients
		2	4			ð6-41Þ
		1	2

is singular.

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To select one from among the infinite number of solution sets, we must have an additional independent nonhomogeneous equation. If the additional equation is

x þ y ¼ 1

ð6-42Þ

the solution set f2; 1g satisfies all three equations [one of the pair (6-40) is superfluous] and is the unique solution set for the homogeneous linear simultaneous equation pair plus the additional equation.

In what immediately follows, we will obtain eigenvalues E1 and E2 for

^ c ¼ c p

H Ei from the simultaneous equation set (6-38). Each eigenvalue gives a - electron energy for the model we used to generate the secular equation set. In the next chapter, we shall apply an additional equation of constraint on the minimization parameters fa1; a2g so as to obtain their unique solution set.

The Secular Matrix

The coefficient matrix of the normal equations (6-38) for ethylene is

a E	a	b	E	ð	6-43	Þ
b

By the criterion of Exercise 2-9, E is an eigenvalue of the matrix in a and b. There are two secular equations in two unknowns for ethylene. For a system with n conjugated sp2 carbon atoms, there will be n secular equations leading to n eigenvalues Ei. The family of Ei values is sometimes called the spectrum of energies. Each secular equation yields a new eigenvalue and a new eigenvector (see Chapter 7).

If we divide each element of the secular matrix by b and perform the substitution

x ¼ a Ei=b, we get
	1	x			ð6-44Þ
	x	1
as the coefficient matrix of the equation set
1		x	a2	¼ 0	ð6-45Þ
	x	1	a1

For the equation set to be linearly dependent, the secular determinant must be zero

	x 1		¼ 0	ð6-46Þ
	1 x

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Expanding the determinant,

x2 1 ¼ 0

so that
x ¼ 1	ð6-47Þ

There are n ¼ 2 roots of the polynomial, one for each eigenvalue in the Ei spectrum.

We are free to pick a reference point of energy once, but only once, for each system. Let us choose the reference point a. We have obtained the energy eigenvalues of the p bond in ethylene as one b greater than a (antibonding) and one b lower than a (bonding) (Fig. 6-3).

π

	α	α	Figure 6-3 The Energy Spectrum of
			Ethylene. The p orbital is bonding, and
	π
			the p* orbital is antibonding.

Finding Eigenvalues by Diagonalization

If we drop x from the secular matrix

x 1

1x

we get
	1	0		ð6-48Þ
	0	1

which has the eigenvalues 1, as we found by expanding the secular determinant [Eq. (6-47)]. If, using an equivalent method, we diagonalize matrix (6-48), the eigenvalues can be read directly from the principal diagonal of the diagonalized matrix

	0	1		ð		Þ
	1	0			6-49a
or
	0	1		ð6-49bÞ
	1	0

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where the order of the roots on the principal diagonal depends on the method of diagonalization but the eigenvalues do not. If we can diagonalize a matrix comparable to matrix (6-48) by deleting x from any secular matrix, we shall have obtained the eigenvalues for the corresponding p electron system, in units of b, relative to an arbitrary energy a. Diagonalization does not change the eigenvalues.

Mathcad	:¼		0			:¼		1
		1					0
A		0	1	B		Þ ¼	1	0
ð Þ ¼		1		ð	B		1
eigenvals A		1		eigenvals			1

By substituting back into the definition of x, we get the solution set for the energy spectrum Ei. In ethylene there are two elements on the diagonal, x11 and x22, leading to E1 and E2. In larger conjugated p systems, there will be more.

If ‘‘dropping x,’’ as is usually said, sounds a little arbitrary to you (it does to me, too), what we are really doing is concentrating on one term of a sum

1		x		¼	0	x	þ 1	0		ð6-50Þ
	x	1			x	0	0	1
Diagonalization (the x matrix is already diagonal) yields
0		x		þ	0		1 ¼	0	x þ 1	ð		Þ
	x 0				1		0	x 1	0		6-51
that is, the roots of the secular matrix are
x		1 ¼ 0								ð	6-52	Þ
x	¼		1

Polynomial root finding, as in the previous section, has some technical pitfalls that one would like to avoid. It is easier to write reliable software for matrix diagonalization (QMOBAS, TMOBAS) than it is for polynomial root finding; hence, diagonalization is the method of choice for Huckel calculations.

Rotation Matrices

If we premultiply and postmultiply the matrix

01

10

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by the matrix

	cos y		sin y		ð6-53Þ
	sin y			cos y
where y ¼ 45 , the result is
	0	1			ð6-54Þ
	1	0

which is the original matrix rotated one-eighth turn or 45 , with a sign change in the second row. What we are really doing is rotating the coordinate system that we have arbitrarily imposed on the wave (vector) function c in a way that is similar to the coordinate rotation we used to discover the principal axes of an ellipse (see

What’s Going on Here?, Chapter 2).

The premultiplying and postmultiplying matrix is often called a rotation matrix R. The rotation matrix

		cos y		sin y
R	¼				ð	6-55	Þ
		sin y		cos y
is widely used to do the same thing as rotation matrix (6-53). It yields
	0	1			ð	6-56	Þ
	1	0

By rotating it through the proper angle, we have diagonalized matrix (6-48). Diagonalization yields the solution set x ¼ ða EÞ=b ¼ f 1; 1g. Multiplying by b,

a E ¼ b	ð6-57aÞ
or
E ¼ a b	ð6-57bÞ

which leads to the two-level energy spectrum for ethylene as shown in Fig. 6-3.

Generalization

The advantage of the method just described is that it can be generalized to molecules of any size. Setting up quite complicated secular matrices can be reduced to a simple recipe. A computer scheme can be used to diagonalize the resulting matrices by an iterative series of rotations.

The dimension of the matrix is the number of atoms in the p conjugated system. Let us take the three-carbon system allyl as our next step. Concentrate on the end

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atom in the system and write an x in the i ¼ j position for that atom. Follow this by writing a 1 in the position corresponding to any atom attached to the x atom. For any atom not attached to x, enter a zero. For allyl, the x goes into the 1, 1 position

C C C

"

This leads to the top row of the secular matrix

x 1 0

Concentrating on the second atom in the allyl chain leads to the row 1 x 1, and concentrating on the third atom gives 0 1 x.

The full allyl matrix is

01

x 1 0

@	1	x	1	ð	6-58	Þ
	0	1	x A

The zeros in the 1,3 and 3,1 positions correspond physically to the assumption that there is no interaction between p electrons of atoms that are not neighbors, a standard assumption of Huckel theory.

If we had been interested in the cyclopropenyl system,

C

C C

we would have been led to the matrix

01

x 1 1

@	1	x	1	ð	6-59	Þ
	1	1	x A

Butadiene,

C C

C C

yields

01

	x	1	0	0
B	1	x	1	0		ð		Þ
	0	0	1	x C			6-60
B	0	1	x	1	C
@					A

and so on. We have moved away from notation involving localized double bonds as in ethylene, and we are working from a picture of p bonds delocalized in some way

HUCKEL MOLECULAR ORBITAL THEORY I: EIGENVALUES

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over a carbon atom framework involving all carbon atoms in the conjugated system. We speak of the entire conjugated system as, for example, the butadienyl system, meaning butadiene and any ions or free radicals that can be derived from it without moving any carbon atoms. In the next chapter we shall be more specific about how the electrons are delocalized over the carbon atom framework.

The rotation matrix R must also be given in general form. If the preand postmultiplying matrix is contained as a block within a larger matrix containing only ones on the principal diagonal and zeros elsewhere (aside from the rotation block), the corresponding block of the operand matrix is rotated. Elements outside the rotation block are changed, too. For example,

0 sin y			cos y		0	1 0	1	0	1		1	0 sin y		cos y	0	1
B	cos y		sin y		0		0	1	0				cos y	sin y	0
	0		0		1 C B		0	1	0 C B				0	0	1 C
@		0				A @					A @					A
			0		1			0:707		1							6-61
		B	1		0		0:707			C						ð Þ
	¼		0:707		0:707			0
		@								A

where y was once again taken as 45 . The rotation matrix can be made as large as necessary to conform with any operand matrix. The rotation block may be placed anywhere on the principal diagonal of the rotation matrix R. The allyl matrix has not been diagonalized by Eq. (6-61), only part of it has.

The Jacobi Method

The Jacobi method is probably the simplest diagonalization method that is well adapted to computers. It is limited to real symmetric matrices, but that is the only kind we will get by the formula for generating simple Huckel molecular orbital method (HMO) matrices just described. A rotation matrix is defined, for example,

0

0

cos y

sin y

0

etc

:

1

B

1

0

0

0

C

R

¼

0

0 y 0

y

1

ð

6-62

Þ

B

0

sin

cos

0

C

B etc:

1

C

B

C

B

etc:

C

B

C

@

A

so as to ‘‘attack’’ a block of elements of the operand matrix. In the case of rotation matrix (6-62), the block with a22 and a33 on the principal diagonal is attacked.

Now,

2aij	ð6-63Þ
tan 2y ¼ aii ajj

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where aij denotes the ij element in matrix A and aii and ajj are on the principal diagonal. (The double subscript on the elements aij distinguish them from the solution set faig.) Initially, element aij is adjacent to element aii and above element ajj as it is in the butadienyl system, but this will not be true later in the diagonalization and is not necessary even at the outset. Element aij can be off the tridiagonal as in the cyclopropenyl matrix, in which case the rotation matrix would be

01

cos y	0	sin y
0	1	0
@ sin y	0	cos y A

The matrix equation

RAR ¼ A0

ð6-64Þ

generates a matrix A0 that is similar to A (see section on the transformation matrix in Chapter 2) but has had elements aij and aji reduced to zero. The eigenvalues of A are proportional to the lengths of the corresponding eigenvectors, and orthogonal transformations preserve the lengths of vectors (Chapter 2), so similar matrices have the same eigenvalues.

The good news is that any aij and aji elements not on the principal diagonal can be converted to zero by choosing the right R matrix. The bad news is that each successive RAR multiplication destroys all zeros previously gained, replacing them with elements that are not zero but are smaller than their previous value. Thus the RAR multiplication must be carried out a number of times that is not just equal to one-half the number of nonzero off-diagonal elements, but is very large, strictly speaking, infinite. The sum of the off-diagonal elements cannot be set equal to zero by the Jacobi method, but it can be made to converge on zero. The Jacobi method is an iterative method.

Let us follow the first few iterations for the allyl system by hand calculations. We subtract the matrix xI from the HMO matrix to obtain the matrix we wish to diagonalize, just as we did with ethylene. With the rotation block in the upper left corner of the R matrix (we are attacking a12 and a21), we wish to find

0			1
	0	1	0
R @	1	0	1
	0	1	0 A R
First,
tan2y ¼			2aij	¼ 1
		aii ajj

y ¼ 45

1

sin y ¼ cos y ¼ p ¼ 0:7071 2

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193

By the simple HMO procedure, it is always true that sin y ¼ cos y ¼ 0:7071 on the first iteration. Now, to eliminate a12,

RAR ¼ A0

ð6-65Þ

but this is just the multiplication we used as an illustration in the last section. We know that the result is matrix (6-61).

0	0	1	0:707	1
@	1	0	0:707	A
	0:707	0:707	0

Because both matrix A and the transformation are symmetrical, reducing the a12 element to zero also reduces a21 to zero. We have gained the zeros we wanted, but we have sacrificed the zeros we had in the 1,3 and 3,1 positions. Other than those eliminated, the off-diagonal elements are no longer zero but they are less than one. Attacking the a13 ¼ a31 ¼ 0:7071 element produces

A00 ¼	0	0::325	: 1	0:628	1
	@	1 37	0 325	0	A
		0	0:628	0:37

and so on for further iterations of the method. Again, the zeros previously gained are lost, but they are replaced by nonzero elements that are less than 0.7071. Nine iterations yield

A000000000 ¼	0		0:	41	0	0	1
	@		1		0	0	A
			0		0	1:41

where elements that are negligibly small, say 10 7, are recorded as zero. The energy levels or eigenvalues for the three-carbon allyl model are

x ¼ 1:41; 0; 1:41

ð6-66Þ

The order of the roots as generated by diagonalization is dependent on the algorithm, as are some of the intermediate matrices generated in the diagonalization procedure. Programs are written to be ‘‘opportunistic,’’ that is, to seek a quick means of conversion on the eigenvalues, and the strategy chosen may differ from one program to the next. Many programs, including the one to be described in the next section, have a separate subroutine at the end that takes the eigenvalues in whatever order they are produced by diagonalization and orders them, lowest to highest or vice versa.

In conclusion of this section, it is remarkable that molecular orbitals are never really used in Huckel theory, that is, the integrals a and b are not evaluated. Huckel