Gallup G.A. - Valence Bond Methods, Theory and applications (CUP, 2002)

16

Interaction of molecular fragments

In	previous chapters	we have repeatedly emphasized that the	principal difficulty
in	calculating the	dissociation energy of a bond is the	correct treatment of the

change in electron correlation as the bond distance changes. This observation also applies to reactions where bonds are both formed and broken. In many important cases, however, the particular atom–atom distances that change significantly during

a reaction are relatively few in number, and a method for accurately treating the correlation in only those “bonds” would have a clear advantage in efficiency. The

MCVB method provides a method for targeting certain bonds to treat the correlation
in them as well as possible. We call this procedure		targeted correlation, and in this
chapter we give examples using it. The SCVB method could also be used in this
context.
In our previous work we have used SCF solutions of the atoms as the ingredients
of the	n -electron VB basis functions. With targeted correlation we go one step up

and use SCF solutions of molecular fragments as the ingredients. As the name implies, this must be tailored to the specific example and must be done with a careful eye to the basic chemistry and physics of the situation at hand.

16.1 Methylene, ethylene, and cyclopropane

In this section we consider some molecules that can be viewed as consisting of methylene radicals in some combination. Earlier publications[39, 66] have covered some of the aspects of the subjects covered here. These earlier studies used an STO3G minimal basis, and provide information to make comparisons of results

with the 6-31G results that are presented here. We will describe the minimal basis results more completely in a later section. Here, however, we make one comment

concerning the way one must handle these different bases. When using minimal bases with targeted correlation qualitatively reasonable results are obtained, but this is, in part, due to the less satisfactory representation of the fragments. When we use

214

16.1 Methylene, ethylene, and cyclopr

opane

215

a larger basis, the fragment is better described, but the orbitals are not so well conditioned to the molecules that one wishes to construct from them. Therefore, when we

use a 6-31G basis it is necessary to allow the open orbitals to breathe as distances change, as suggested by Hiberty[44]. We will discuss the methylene biradical first.

		16.1.1 The methylene biradical
The structure of CH		2 was discussed by a completely MCVB treatment in Chapter 15.
Here we look at it from an ROHF point of view. The structure of CH					2 was uncertain
for a number of years, but it is now known that the ground state is triplet with a bent
geometry in		C 2v symmetry. Conventions dictate that CH			2 be oriented with the	C 2-
and z -axes coincident and the molecule in the					y – z plane. Consequently the ground
state is	3 B 1, and the MO configuration is
		1a 122a 121b 223 a 11b 1.
The first excited state is the singlet configuration
		1a 22a	21b	23 a	2,
		1	1	2	1
which has	1 A	1 symmetry. The SCF energies for these are rather too far apart since
there is more electron correlation in the singlet coupling. We shall be able to interpret
our results for ethylene and cyclopropane in terms of these states of the methylene
biradical.

					16.1.2	Ethylene
Our treatments of ethylene are all carried out with two methylene fragments that
have the 1	a 22a	2	1b 2	parts of both of their configurations doubly occupied in all VB
	1	1	2
structures used. The 12 electrons involved can be placed in the core as described in
Chapter 9, which means that there are only four electrons, those for the C—C								σ and
π bonds, that are in the MCVB treatment. For simplicity we shall rename the other
two methylene orbitals				σi and	πi , where	i = 1, 2 for the two ends of the molecule.
The Weyl dimension formula tells us that there are 20 linearly independent tableaux
from four electrons distributed in four orbitals. When we use							D 2h	symmetry, how-
ever, only 12 of them are involved in eight						1 A	1 functions.
As indicated above, the				σi	and πi orbitals are not the “raw” orbitals coming out
of the ROHF treatment of methylene, but linear combinations of the occupied and
selected virtual orbitals of that treatment, which provides the breathing adjustment.
Specifically, we use
			σi	= c 13 a 1 + c 24a 1 + c 3 5a 1 + c 46 a 1 + c 57a 1,				(16.1)
			πi	= d 11b 1 + d 22b 1 + d 3 3b			1,	(16.2)

216	16 Interaction of molecular fragments

	14
	12
		21A	1
	10
(eV)	8
Energy	6
	4	11A	1
	2
	0
	1	2	3	4	5	6	7	8	9	10

C — C bond length (A)

Figure 16.1.

Dissociation curve for the double bond in CH

2=CH 2.

and optimize the six independent (each orbital is normalized) parameters at each

C—C distance. This includes all of the

b 1 virtual orbitals but omits the highest three

a 1 virtual orbitals. These latter are mostly involved with the 1

s function of the basis

and will not influence bonding significantly. Figure 16.1 shows the ground and first

excited singlet states of ethylene as a function of the C—C distance. The molecule

is held in a plane and possesses

D 2h

symmetry at

all distances. The H—C—H

angle, as determined from SCF minimizations, changes by about a degree in this

transformation, but this nicety was not included, the angle being held at the ethylene

value for all distances.

At

R CC

= ∞, the

ground

state wave function is

particularly

simple in terms of

standard tableaux functions,

= ∞)=

π1

,

0 (R

π2

(16.3)

σ1

σ2

where we

assume

the tableau

symbol

includes its

normalization

constant. This

is easily interpreted as two triplet systems coupled to singlet. In terms of HLSP

functions the results are not so simple. We have

R

− 0 .577 35

σ1

R

0 (R = ∞)= 0 .577 35 π1

π2

π2

,

(16.4)

σ1

σ2

σ2

π1

since these do not represent the triplet states so easily. In neither case are there any ionic terms, of course.

16.1 Methylene, ethylene, and cyclopr opane 217

Table 16.1.

The principal terms in the ground state wave function for R

CC at the

energy minimum. The two sorts of tableaux are given.

1

2

3

4

Standard

Num.

1

2

2

1

tableaux

σ1

σ2

σ2

σ2

π2

π2

σ1

π1

functions

Tab.

π1

π2

π1

π2

σ1

σ2

σ2

π2

C i (min)

0.536 47

0.148 46

−0.128 37

−0.124 77

HLSP

Num.

1

2

2

2

functions

Tab.

σ1

σ2

σ2

σ2

π1

π1

σ1

σ1

π1

π2 R

π1

π2 R

σ1

σ2 R

π2

π2 R

C

(min)

0.473 51

0.148 46

0.128 37

0.121 06

i

−

−

The first excited state wave function is also more complicated at

R

CC

= ∞. In

terms of standard tableaux functions it is

1(R

= ∞)=

0.912 69

σ1

σ1

− 0.399 26

σ1

σ1

σ2

σ2

π2

π2

− 0.399 26

σ2

σ2

+ 0.087 09

π1

π1

.

(16.5)

π1

π1

π2

π2

The first term is the combination of two

1 A

1 methylenes, and

the

others provide

some electron correlation in these two structures. Since Eq. (16.5) has only doubly

occupied tableaux, the HLSP functions are the same.

We show the ground state wave function at

R min

in terms of standard

tableaux

functions and HLSP functions in Table 16.1. We see that the representation of the

wave function is quite similar in the two different ways. Considering the HLSP

functions first, we note that the principal term represents two electron pair bonds,

one σ and one

π . The next two are ionic structures contributing to delocalization,

and the fourth is a nonionic contribution to delocalization.

The standard tableaux function representation is similar. The principal term is

the same as the only term at

R = ∞, and together with the fourth term (the other

standard tableau of the constellation) represents the two electron pair bonds of the double bond. The second and third terms are the same as those in the HLSP function representation and even have the same coefficients, since there is only one function

of this sort.

When we make a similar analysis of the terms in the wave function for the first excited state, more ambiguous results are obtained. These are shown in Table 16.2. For

both standard tableaux functions and HLSP functions the principal structure is the same as that in the ground state. Higher terms are of an opposite sign, which provides

the necessary orthogonality, but the character of the wave function is not very clear

218 16 Interaction of molecular fragments

Table 16.2.

The principal terms in the first excited state wave function

for

R

CC

at the energy minimum. The two sorts of tableaux are given.

1

2

3

4

Standard

Num.

1

2

2

1

tableaux

σ1

σ2

π2

π2

σ1

σ1

σ1

σ1

functions

Tab.

1

π1

π2

σ1

σ2

π2

π2

π1

π2

C

i

0.711 57

0.478 12

0.337 33

0.216 83

HLSP

Num.

1

2

2

2

functions

Tab.

σ1

σ2

π1

π1

σ1

σ1

σ1

σ1

π1

π2

σ1

σ2

π2

π2

π1

π2

C

i1

0.578 01 R

0.478 12 R

0.337 33

R

0.216 83

R

from these terms. We give this example, because we will contrast it with an excited

state wave function of an unambiguous sort when we discuss cyclopropane. There is

one more point that should be discussed before we go on to cyclopropane, however.

The 1

a 122a 121b 22 orbitals on one of the two CH

2 fragments are not orthogonal to

those on the other CH

2,

but this does not cause the core valence

separation

any

problems. It does, however, represent a repulsion: that which is normally expected

between closed-shell systems. In this case the overlap and the repulsion are small.

During the calculations of the core matrix elements a measure of the overlap is

computed. This number is exactly 1.0 if there is zero overlap between the fragment

core orbitals. As more overlap appears the number rises and can be as high as 100.

For ethylene we never press the CH

2

fragments close enough together to

reach

numbers higher than about 1.02. This is quite small and is the reason we know that

the core repulsion is small in this system.

16.1.3

Cyclopropane with a 6-31G

basis

We examine the two lowest singlet states of cyclopropane as one of the CH

2 groups

is pulled away from the other two. Figure 16.2 shows the basic arrangement of the

molecule with the three C atoms in the

x– y plane. The C atom on the right is on the

y -axis, and

R 1 is its distance to the midpoint of the other two Cs.

R 2 is the distance

between the two Cs that will become part of ethylene and

φ is the angle out of

planarity. We have labeled the C atoms 1, 2, and 3 to identify the three different

methylenes for designating orbitals.

The wave function is an extension of the one we used for the dissociation of

ethylene. We now have 18 electrons in nine core orbitals, and six electrons in the

three

σ and

three

π orbitals that will make up

the C—C

bonds. As

before, the

valence orbitals are allowed to breathe (see Eqs. (16.1) and (16.2) for the linear

combinations) as

the

system changes. According

to the Weyl

dimension

formula

16.1 Methylene, ethylene, and cyclopr			opane			219
φ
H
C2
R2		R1	C1		H
C3
H
φ
Figure 16.2. The geometric arrangement of the atoms of cyclopropane during the dissoci-				2v geometry as		R 1, R 2, and
ation to ethylene and methylene. The system is maintained in			C
φ change.

	14
	12
	10	21A	1
(eV)	8
Energy	6
	4
	2	1	1
		1 A
	0
	1	2	3	4	5	6
			R	1(Å)

Figure 16.3.		The two lowest			1 A	1 states during the dissociation of cyclopropane along the
C 2H 4 relaxed path.			The dashed lines, indicating the diabatic energies, were not computed
but have been added merely to guide the eye.
there are 175 different structures possible, but in this case there are only 173 of
them involved in 92				1 A 1 (C	2v ) symmetry functions. The core overlap criterion never
becomes larger than 1.021 for these calculations.
In Fig. 16.3 we show the two lowest						1 A 1 states as a function of	R 1 for optimum
values of the		R 2 and		φ parameters, and in Fig. 16.4 we show the path by giving
R 2 and	φ as functions of			R	1. Qualitatively, the two energy curves have the classic
appearance	of	an	avoided crossing			between two diabatic states. The dashed lines
in Fig. 16.3 are not computed, but have been added to guide the eye.
Although we have been speaking of our process as the dissociation of cy-
clopropane,	it is simpler		to	discuss it	as	if proceeding from the other direction.

220	16 Interaction of molecular fragments

	30									2.0
	25		← φ
										1.8
	20
	15									1.6	(Å)
(degrees)
											2
	10						R	2 →			R
φ										1.4
	5
										1.2
	0
	− 5									1.0
	1.0	1.5	2.0	2.5		3.0	3.5	4.0	4.5	5.0
					R	1	(Å)
Figure 16.4.		The variation of		R	2 and	φ during the dissociation of cyclopropane.

Considering the dissociated system, we see that the geometry has relaxed to that of an ethylene molecule with a methylene radical at some distance. Our treatment allows methylene to have only two electrons in open shells so it must be in either a

singlet or a triplet state. Since the overall system is in a singlet state, the ethylene portion must also be either singlet or triplet, respectively, to match. At the same time ethylene is singlet in its ground state and triplet in its first excited state with a fairly large excitation energy, while methylene is triplet in its ground state and singlet in its first excited state with a relatively small excitation energy. These facts

tell us that, at long distances, the lower		1 A 1 state is a combination of ground state
ethylene and singlet methylene. The Rumer tableau that corresponds to this case is
	σ2	σ3		,
	σ1	σ1
	π2	π3	R
where the subscripts identify the particular	methylene	fragment as		given in
Fig. 16.2. There is, of course, another Rumer structure corresponding to this orbital
set. Table 16.3 gives the coefficients in the wave function.
At long distances the first excited	1 A	1 state, on the other hand, is a combination

of triplet ethylene and ground state (triplet) methylene. The important tableaux in the wave function are shown in Table 16.4, and in this case the principal tableau of

16.1 Methylene, ethylene, and cyclopr opane 221

Table 16.3.

The leading tableaux for the ground state wave function

of C

2H4 + CH

2 at infinite separation.

1

2

3

4

Standard

Num.

1

1

2

2

tableaux

σ1

σ1

π1

π1

σ1

σ1

σ1

σ1

functions

Tab.

σ2

σ3

σ2

σ3

σ2

σ2

σ2

π2

C i (∞)

π2

π3

π2

π3

π2

π3

σ3

π3

0.529 89

−0.156 11

0.146 58

−0.128 41

HLSP

Num.

1

2

1

2

functions

σ1

σ1

σ1

σ1

π1

π1

σ1

σ1

Tab.

σ2

σ3

σ2

σ2

σ2

σ3

π2

π2

π2

π3

R

π2

π3

R

π2

π3

R

σ2

σ3

R

C i (∞)

0.463 18

0.146 58

−0.136 47

−0.120 81

Table 16.4.

The leading tableaux for the first excited state wave function

of C

2H4 + CH

2 at infinite separation.

1

2

3

4

Standard

Num.

1

1

2

2

tableaux

σ1

σ2

σ1

σ3

σ3

σ3

σ1

σ3

functions

Tab.

σ3

π2

σ2

π2

σ1

π2

σ2

π1

C i (∞)

π1

π3

π1

π3

π1

π3

π2

π3

0.414 52

0.325 77

−0.206 49

0.096

43

HLSP

Num.

1

2

1

2

functions

σ2

σ

σ2

σ

σ3

σ

σ3

σ

Tab.

π1

π32

σ1

π31

π1

π32

σ1

π31

σ1

π3

R

π2

π3

R

σ1

π3

R

π2

π3

R

C i (∞)

0.801 48

0.383 16

0.245 93

0.133 59

the state is more easily written

σ3

1 in terms of standard tableaux functions as

π2

π2

σ1

,

σ1

σ2

σ2

σ3

π1

π3

= − π3

π1

where there are, of course, four more standard tableaux that could be written. We may actually use the tableau shapes to see that this is the correct interpretation of

1In this book the entries in tables like Table 16.4 are generated semiautomatically from computer printout. Although this almost completely eliminates the dangers of misprints the computer programs do not always arrange the tableaux in the most convenient way for the discussion. In this case we use the properties of standard tableaux functions to make the transformation used.

222 16 Interaction of molecular fragments

Table 16.5. The leading tableaux for the ground state wave function of C

3 H6 at

the equilibrium geometry.

1

2

3

4

Standard

Num.

1

1

1

1

tableaux

σ1

σ2

σ1

σ

σ1

σ2

π1

π1

functions

Tab.

σ3

π2

σ2

π31

σ3

π1

σ2

σ3

π1

π3

π2

π3

π2

π3

π2

π3

C i (min

)

0.134 41

0.130 92

−0.123 14

0.102 56

HLSP

Num.

1

1

1

2

function

σ2

σ3

σ1

σ2

π1

π1

π3

π3

Tab.

π1

π2

σ3

π1

σ2

σ3

σ1

σ2

σ1

π3

R

π2

π3

R

π2

π3

R

π1

π2

R

C i (min

)

0.132 70

−0.125 19

0.086 67

0.085 38

the states of the separate pieces of our system. The last tableau given above can also be written symbolically as

π2	σ	3		σ1		,
σ2			+ π1
π3
where the two triplet tableaux can fit together to form the earlier singlet shape. We
have not emphasized this sort of combining of tableaux in our earlier work, but it
is particularly useful for systems in asymptotic regions. There are more structures
for this state and set of orbitals than for the lower one, because this set must also
represent a still higher coupling of ethylene and	methylene, both			in a			1 B 1 state.
We do not show the energy curve for the state that goes asymptotically to that
coupling.
The wave functions for the ground and first excited							1 A 1 states for the cyclo-
propane equilibrium geometry are shown in Tables 16.5 and 16.6. In the case of
the ground state either the principal standard tableaux function or HLSP function
can be transformed as follows:		π2				σ1
σ3	π2						,
σ1	σ2			σ2		σ3
π1	π3	= − π3				π1

π1	π32	π2		σ1 .
σ2	σ		σ2	σ3
σ1	π3	R	= π3	π1 R

16.1 Methylene, ethylene, and cyclopr opane 223

Table 16.6.

The leading tableaux for the first excited state wave function of C

3 H6

at the equilibrium geometry.

1

2

3

4

Standard

Num.

1

1

1

1

tableaux

π1

π1

σ1

σ

σ1

σ

σ1

σ2

functions

Tab.

σ2

σ3

σ2

π32

σ2

π31

σ3

π1

π2

π3

π1

π3

π2

π3

π2

π3

C i (min)

0.205 06

−0.182 87

0.176 27

−0.165 80

HLSP

Num.

1

1

1

1

functions

σ2

σ

π1

π1

σ1

σ2

σ2

σ

Tab.

π1

π32

σ2

σ3

σ3

π1

σ1

π31

σ1

π3

R

π2

π3

R

π2

π3

R

π2

π3

R

C i (min)

0.182 33

−0.173 96

0.168 55

0.167 47

The first of these is the same as the transformation above for the first excited

asymptotic wave function. Thus, at the minimum

geometry we see that the leading

∞, and there

term in the wave function is the same as that for the first excited state at

has been a cross-over in the character of the wave function for the two geometries.

The leading coefficients are rather small for these functions. This is in

part

because the orbital set in terms of which we have expressed the

functions

is

not

the most felicitous. We have used

σi and

πi

relating to the local geometry of each

methylene. Alternatively, we can form hybrid orbitals

h i 1 = N (σi + πi ), h i 2 = N (σi − πi ),

which, in each case, are directed towards a neighboring methylene. The signs of the orbitals are such that these combinations yield

h 12 ↔ h 21, h 22 ↔ h 31 , h 32 ↔ h 11,

as the pairs that overlap most strongly. Table 16.7 shows the HLSP function tableaux for the ground state in terms of these hybrids. This representation gives little clue

as to the asymptotic state this might be connected with, but does show a rather conventional picture of cyclopropane as having three electron pair bonds holding

the ring together. There is also the expected mix of covalent and ionic functions. When we get to the first excited state at the geometry of the energy minimum

(Table 16.6), it is seen that most important tableaux in the wave function in terms