A COMPUTATIONAL STUDY OF ATOMIC CHARGE
OF BORON TRIHALIDES AND RELATED COMPOUNDS
by
Bei Liu
B.S., Nanjing Agriculture College, 1990
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Chemistry
1998
1998 by Bei Liu
All rights reserved.
This thesis for the Master of Science
degree by
Bei Liu
has been approved
by
Doris Kimbrough
Bei Liu(M.S., Chemistry)
A Computational Study of Atomic Charge of Boron Trihalides and Related Com-
pounds
Thesis directed by Professor Robert Damrauer
ABSTRACT
The purpose of this study is to investigate the atomic charge of a series
molecules. The calculation has been carried out at MP2/6-3lG*//MP2/6-311G**
level by Gaussian 94 packed in digit computer. These compounds include the
halides of second row B, C, N and third row P. Some ir substituent of these
atoms have been brought into calculation such as -OH, -SH, -NH2, -PH2, in
order to give a complete comparison of the effect on atomic charge from elec-
tronegativity and 7T back donation.
As one of the most recommended algorithm, Baders AIM method has been
chosen for the calculation and the quality has been examined. High level of ba-
sis sets ensure the best performance of program. Most of our molecules have
been previously reported and the results from our calculation have been regarded
consistent with these studies carried by different methods such as Mulliken, Na-
ture Bond Orbital. Our research also investigates some new compounds which
IV
have not yet been found to exist. The study provided a chance to predict the
stablization energy and atomic charge of these possible compounds and hence
their Lewis acidity or basicity also could be estimated based on these parameters
derived fronTc'alculation.
Among our results AD3(D = NH2, PH2, OH, SH) has a good agreement
with the electronegativity of substituent. However, the halides(F, Cl, Br) of
all these atoms have some discrepancy, most anomalous happened on bromine,
suggesting the program still needs to be improved for large atoms.
This abstract accurately represents the content of the candidates thesis. I rec-
ommend its publication.
Signed
Robert Damrauer
v
ACKNOWLEDGEMENTS
I would like to take this opportunity to thank my advisor, Professor Robert
Damaurer, for the guidance and encouragement that he provided during all phases
of the preparation of this thesis. Throughout my Master studies, he provided
exceptional insight and guidance. I significantly benefited from the countless and
fruitful discussions that I had with him.
My sincerest thanks also go to Professor Larry Anderson, who provided me
with close supervision. His outstanding knowledge of physical chemistry and his
continuous support, encouragement and interest greatly enhanced the quality of
this work.
Thanks also go to Dr. Doris Kimbrough, for her contribution to my develop-
ment.
Special acknowledgments goes to my husband, Zhining, for his understanding
and support during my study.
DEDICATION
To Grandma
CONTENT
Figures...................................................... xi
Tables.......................................................xiii
CHAPTER
1. INTRODUCTION.............................................. 1
2. THEORY BACKGROUND......................................... 6
ab initio Quantum Chemistry Theory........................ 6
Schrodinger Equation...................................... 8
Solving the Electronic Schrodinger Equation.............. 11
Constructing a Many-Electron Wavefunction from Molecular
Orbital............................................ 11
Molecular Orbitals Theory: the LCAO/MO Method......... 13
Basis Functions....................................... 14
Hartree-Fock Theory................................... 16
Population Analysis..................................... 21
Moller-Perturbation Theory.............................. 23
Quality of ab initio Results ............................ 25
Geometry.............................................. 26
vm
Vibrational Frequencies
27
Energies............................................. 27
Atomic Charge........................................ 28
3. COMPUTATIONAL METHODS................................... 29
The Schematic Route..................................... 29
Geometries ............................................. 32
Basis Set............................................... 33
Energy.................................................. 33
Frequency Calculation................................... 34
Atomic Charge........................................... 34
4. RESULT AND DISCUSSION................................... 36
Trihalide and Hydrogen Compounds AX3.................... 38
jBX3................................................. 38
CX4.................................................. 42
NX3.................................................. 45
P X3................................................. 48
AY3, Y = OH, SH, NH2, PH2............................... 51
BY3................................................. 52
CY4.................................................. 54
NY3.................................................. 56
PY3.................................................. 56
IX
Phosphine Oxides POXz{X=H, F, Cl, Br).......... 58
5. CONCLUSION..................................... 64
BIBLIOGRAPHY ...................................... 69
FIGURES
2.1 Electron configuration diagram for (a) (0ia)('0i/3)('02Â£*)(V,2/3)(V,3Q:)
and (b) ............................ 20
3.1 Logical Structure of ab initio molecular orbital program......... 31
4.1 The optimized structures and atomic charges of BX$............... 40
4.2 The atomic charge of CX4........................................... 44
4.3 The optimized structures and atomic charge for NX3............... 47
4.4 The optimized structures and atomic charge for PX3............... 49
4.5 The optimized structures and atomic charges of BY3. a: Y=NH2,
b: Y=PH2, c: Y=OH, d: Y=SH....................................... 53
4.6 The optimized structures and atomic charges of CY4. a: Y=NH2,
6: Y=PH2, c: Y=OH, d: Y=SH....................................... 55
4.7 The optimized structures and atomic charges of NY3. a: Y=NH2\
b: Y=PH2-, c: Y=OH- d: Y=SH...................................... 57
4.8 The optimized structures and atomic charges of PY3. a: Y=NH2;
b: Y=PH2; c: Y=OH- d: Y=SH....................................... 59
4.9 The optimized structures and atomic charges of POX3............... 61
xi
5.1 Summary of atomic charge at A in AX3 and CX4. a: X=H, b:
X=F, c: X=Cl.................................................. 65
5.2 The'atomic charge of A in a: A(OH)3(A) and i4(5,if)3(D); b:
N..
A(NH2)3 and A(PH2)3{)........................................ 67
xii
TABLES
4.1 The optimized energy and single point energy(in au) 37
4.2 The bond length(B-X), bond angle(X-B-X), atomic charges(Boron)
and charge capacities of BX3.................................. 39
4.3 The bond length(C-X), bond angle(X-C-X), atomic charge(Carbon)
and Charge Capacities of CX4.................................. 45
4.4 The bond length(N-X), bond angle(X-N-X), atomic charge(nitrogen)
and charge capacities For NX3 ................................. 46
4.5 The bond length(P-X), bond angle(X-P-X), atomic charge(P) of
PXZ............................................................ 50
4.6 The atomic charges of A in AY3................................. 51
4.7 The bond length(P-O), atomic charge(P) of POX3 at MP2/6-
311G**//MP2/6-31G*............................................. 60
xm
CHAPTER 1
INTRODUCTION
Quantitative evaluation and prediction of electron donor-acceptor interactions
has been always an interesting field for organic chemists[l] [2]. Lewis acid boron
trihalides are among those topics which have been studies intensively. It has
been recognized that the Lewis acidity of BX3 increases in the order BF3 <
BCl3 < BBr3. An electronegativity argument would suggest that fluorine, the
most electronegative halogen, ought to leave the B atom in BF3 the most elec-
tron deficient and, hence, the most acidic. A back-donation approach has been
used to explain this paradox[3]. According to this approach, the halogen atoms
in the BX3 molecule can form 7r bonds with the empty 2p orbital on boron, and
these 7r bonds must be broken to make the acceptor orbital available for reaction
with a Lewis base. The small F atom is said to be the best able to form such
7r bonds with the 2p boron orbital (because its smaller 2p orbitals are supposed
to optimally overlap with the 2p boron orbital.). So the BF3 has the strongest 7r
bond to be broken when Lewis bases attack, while BBr3 is the strongest Lewis
acid where its 7r bonding is the weakest because its p orbitals match the boron 2p
ortital poorly. One of the proof supporting this approach can be found in Olahs
1
paper[4], where they showed that the overlap between p(ir) orbitals of carbon
and the halogens decrease as: 2p(7r) 3p(7r) > 2p(7r) 4p(7r) > 2p(7r) 5p(7r).
However, this explanation has been challenged recently. Brinck and coworkers [5]
x.
calculated the atomic charges of BX3(X=F, Cl) and the complexation energies
for the interactions of BX3 with NHz- The atomic charge of boron were in agree-
ment with the electronegativity variation of halogen: thus, boron in BF3 has a
1.491 charge which in BCI3 it is 0.291. Both the Natural Bond Orbital popula-
tion analysis and the Merz-Singh-Kollman method were adopted for calculating
charges from electrostatic potentials. At all levels of theory, boron is much more
positive in BF3 than in BCl3. Furthermore, both population analyses find the
boron 2p7r orbital to be more populated in BCl3 than in BF3. This result does
not support the back-bonding concept, since in that case, the boron charge in
BF3 should be in the opposite direction.
As an alternative explanation, these workers defined the charge capacity, k, to
explain the trend of the acidity. According to their approach, k can be estimated
from the molecular ionization potential I and electron affinity A,
1
I-A
(1.1)
both I and A of boron trihalide can be found from reference[6], and the values
of k from their calculation seem to vary in the same direction as their electron
affinities. Hence, the trends of electron affinities can be considered in terms of
2
the increasing ability of the larger, more polarizable congeners to accommodate
the additional electron.
In Brinck paper[5] researchers try to confirm their point by using the atomic
'-v._
charge as one of the parameters. There are a variety of procedure can be
used from ab initio molecular orbital calculations: Mulliken population analy-
sis and Weinhold-Reed Natural Bond Orbital(NBO) Analysis[7] are two proce-
dures based on the molecular orbitals. Hirshfeld Procedure and Baders Atoms
in Molecules(AIM) method[8] use charge density distribution. Also, charges can
be derived by fitting the electrostatic potential(CHELPG), etc. However, in a re-
cent study, Wiberg and Rablen [9] compared various charge calculation methods
and find that Baders AIM method is the best in representing the polarization of
bonds. A satisfactory representation of charge should be independent of the basis
set used in the calculations, reproduce reasonable trends due to electronegativity
differences, and reproduce the dipole moment of the molecule. While the AIM
method is the best method tested, one should recognize that no single procedure is
ideal for all purposes. Most of the methods had difficulty in giving atomic charges
that could reproduce both the dipole moments and the electrostatic potentials
of a representative set of molecules in a satisfactory manner. The Mulliken pro-
cedure has a strong dependence on the basis set. The Weinhold-Reed NBO
Population Analysis improves on this problem but its charges give a large C H
bond dipole in the opposite sense from that found experimentally. AIM, though
3
strongly recommended by Wiberg and Rablen, is very SIZE dependent when the
molecule is spherically symmetrical, and it will lead a exaggerated charge den-
sity at electronegative atoms[10]. It has been known that atomic charge, unlike
observable electron density p, has never been well-defined. One definition re-
quires that the atoms be spherically symmetrical and nuclear centered, such as
CHELPG, another makes use of more physically reasonable nonspherical atoms,
which used by both NBO and AIM method. It is always arbitrary. The reason
that the AIM method be recommended is it can provide a natural definition of
atomic dipoles and higher terms such as showing that the zero-flux surfaces are
natural boundaries, with many useful properties. It is sensible to try to use them
as definition of atomic charge, and if these are included, the electric moments
and electrostatic potentials of the molecules are well reproduced. Until the newer
methods for assigning atomic charges proposed, NBO and AIM will continue to
be the methods of choice in computational research.
The confusion in various studies arising from using different atomic charge
methods reported and the fact that no study has tried to compute the charges
on bromine atom suggested to us that a careful study of boron and its close
relatives would be an important advance. Since there is no detailed report using
AIM calculations, we decided to carry these out on the second row elements from
boron to nitrogen. Phosphorous has been included too because of its similarity
to nitrogen. These elements have been substituted by H, F, Cl, and Br, and
4
certain ir donor group like OH, SH NH2 and PH2.
After we had undertaken this study and finished a number of aspects of it,
Frenking and coworkers published their paper[ll]. Their report and some of
S.
our work are very similar, for example, the analysis on boron trihalide com-
pounds(from BF3 to BBr3). By investigating of a series cations and neutral
molecules, Frenking and coworkers questioned the Olahs conclusion[4] that the
decreasing order of back-donation: Cl > Br > I. They believe that is in conflict
with earlier quantum chemical, calculations. In order to prove their point, the
whole group 14 have been taken into the calculation. Their Nature Bond Or-
bital(NBO) analysis for cations AXÂ£ (A=C, Si, Ge, Sn, and Pb\ X=F, Cl, Br,
and I), AHÂ£ and the isoelectronic neutral Lewis acids YX3 (Y=B, Al, Ga, In
and TV) and YH2X, has shown the same trend reported by Brinck and coworkers,
where the atomic charge of boron in BF3 to BI3 decrease from 1.48 to -0.45, just
in agreement with the decreasing order of halogens electronegativity. In addition
to that, energetic stablization, hydride affinity, pir population at atom A or Y
were all calculated and the result leading to an opposite fact to the back-donating
approach: instead of decreasing, the pir donation ability of the halogens actually
increase from F to the I.
Unlike the AIM method we carried out, the atomic charge of all these com-
pounds in Frenking and coworkers study were calculated by NBO method. It is
very interesting to compare our result with theirs and the difference from these
5
two methods breaks new ground in this area. Also, even there were other reports
on the relating compounds such as POX3 [12], carbonic acid [13], and P(N 112)3
[14] etc., however, an atomic charge analysis on these compound hasnt been
found yet. The calculation of these pn donating group, combining with the result
of halogen compounds inspire us to pursue a further investigation on the validity
of atomic charge as a parameter for deciding the molecule property, especially
acidity.
6
CHAPTER 2
THEORY BACKGROUND
ab initio Quantum Chemistry Theory
Classical methods in the form of molecular mechanics and dynamics can be used
to model many aspects of molecular structure and dynamics. The classical force
field is basically built on empirical results, averaged over a large number of
molecules. However, many important questions in chemistry can not be sim-
ply addressed by means of these empirical approach, a more fundamental and
general theory must be sought. Quantum Chemistry provides an approach which
can give not only structure or other properties that are derived just from the
potential energy surface, but those particular properties that depend directly on
the electron density distribution, including those non-standard cases for which
molecular mechanics is simply not applicable.
The quantum chemistry theory is based on the postulates of quantum me-
chanics. In simple terms, the system(such as energy, geometry, and many other
properties) is described by a wavefunction which depends on the position and
time coordinates. This wavefunction can be found by solving the Schrodinger
equation. The Hamiltonian operator, which can be considered as the recipe for
6
obtaining the energy associated with a wavefunction describing the positions of
the the nuclei and electrons in the system, are related to the stationary states
of the systenrand their energies by this equation. In practice, the Schrodinger
equation cannot be solved exactly for real system(though it does can do for H2),
so approximations have to be made. This theory is called ab initio when it uses
no empirical information, except for the fundamental constants of nature such as
electrons mass, Plancks constant, etc.
Despite the approximations which are necessary, ab initio theory has the con-
ceptual advantage of generality, and the practical advantage that its success and
failures are more or less predictable. The major disadvantage of ab initio meth-
ods are the heavy demands on computer power. Therefore, it was necessary
to introduce further approximations which bring empirical parameters into the
theoretical model. This is another quantum chemistry approach: the so called
semi-empirical quantum chemical methods. Compared with ab initio calcula-
tions, their reliability is less and their properties are limited by the requirement
for empirical parameters. Except for larger systems, one should apply ab initio
methods, which generally can be used at a very high level.
7
Schrodinger Equation
According to quantum mechanics, the energy and many properties of a stationary
state of a molecule can be obtained by solution of Schrodinger equation:
me,n = EVe,n. (2.1)
A
here, H is the Hamiltonian operator, a differential operator representing the total
energy. It is the sum of three parts: the kinetic energy of the nuclei, the kinetic
energy of the electrons, and the potential energy of the nuclei and electrons:
H = Tn +Te + Ve>n. (2-2)
$ is a wavefunction. It depends on the geometric coordinates of all particles
and also spin coordinates. The square of the wavefunction, P2, or |^2| if it is
complex, is interpreted as a measure of the probability distribution of the particles7
within the molecule. E, is the numerical value of the energy of the state, which
is the energy relative to a state where the nuclei and electrons are infinitely
separated and at rest.
To solve the Schrodinger equation, four approximations are commonly made:
1. Time independence: Since the time independent Schrodinger equation is
more easily solved, we generally consider only the states that are only sta-
tionary in time.
8
2. Nonrelativistic. Since a non-relativistic Schrodinger equation is more easily
solved, it is used except when the velocities of the particles, particularly
electrons, approach the velocity of light.
3. Separation of nuclear motion-The Born-Oppenheimer approximation. The
purpose for this is that the nuclear masses are so much greater than those
of the electrons, that nuclei move much more slowly, hence the electrons
can adjust their distribution to changing nuclear positions rapidly. In
practice, this approximation is usually valid. Named after two scientists,
the Born-Oppenheimer approximation can be formulated quantitatively by
Schrodinger equation for electrons in the field of fixed nuclei:
He(Rn)i^{i'e) = Ee(Rn)y(re). (2.3)
This equation still contains the positions of the nuclei, however not as vari- ^
ables.
4. Orbital approximation. By this way the wavefunction $ of the N electrons
in molecule now can be written as a product of N one-electron wavefunc-
tion if) and these one-electron wavefunctions are the molecular orbitals. As
a result, the electrons are confined to certain regions of space, which usu-
ally represented by an electron configuration diagram such as shown in the
Figure 2.1. The electrons are represented by arrows (| for a, J. for /? ), the
orbitals of lowest energy lies on the bottom of the diagram.
9
In order to simplify expressions and to make the theory independent of the
experimental values of physical constants, atomic units are introduced:
e = 1 charge of electron
m = 1 mass of the electron
ft, = 1 Plancks constant divided by 2 7r
length as Bohr radius eto, defined by:
%2
1 bohr = a0 = - = 0.529A
me1
Energy as hartree,
1 hartree
= 4.3598 x 10 18 J/molecule
oo
627.51 kcal/mol.
Bring these into the Schrodinger equation,
HV
e,n
= EV
e,nj
(2.4)
we can get the sum of kinetic and potential parts of electronic Hamiltonian:
in n N *7
= -;Â£*-Â£Â£
i=l
n -i
/ / I r- | + XI>
t=l A-\ r,'l i
(2.5)
where V is the Laplacian:
d2 d2 d2
V1 o_2 "t" a..2
dx\ dy2 dzf
10
Solving the Electronic Schrodinger Equation
Constructing a Many-Electron Wavefunction from Molecular Orbital
The Hamiltonian contains three terms for an electron: its kinetic energy, the
electron-nuclear attraction, and the repulsion of other electrons, as expressed in
equation 2.5. The repulsion term,
n n I
= (2.6)
i
depends on the coordinates of two electrons at the same time, hence has become
a practical computational problem, because it only works for very small systems.
It is possible to avoid this problem by introducing the independent particle ap-
proximation: the interaction of each electron with all the others is treated in an
average way. If:
71 71 I 71
'Â£% = 'Â£= 'Â£vr (2-7)
t
then the Schrodinger equation which initially depended on the spatial and spin
coordinates of all electrons can be reduced to a set of equation:
71
!,X2,...,Xn) = Ety(xi, X2, ..., xn) (2.8)
i=1
Hi4>i(xi) = FiMXi) = eiMXi) (2-9)
where the operator can be considered as the sum of kinetic and potential terms
and the wavefunctions are called one-electron spin-orbitals which is the
11
product of a molecular orbital and a spin function, and they are used to approx-
imate the full wavefunctions according to the Hartree-Fock theory.
Before going further, a couple of things should be kept in mind: first, the
spin-orbitals should be antisymmetric, this could be done by arranging them in a
determinantal wavefunction which ensure the property of antisymmetry; second,
orbitals need to be normalized and forced orthogonal to each other. With these
characters, a full many-electron molecular orbital wavefunction for the closed-
shell ground state of a molecule with n(even) electrons, doubly occupying n/2
orbitals could be expressed by such a determinant:
^i(l)(l) V>i(l)/?(1) ^2(l)a(l) Vty2(l)/?(1)
V,i(2)a(2) ^(2)0(2) 02(2)a(2) 0n/2(2)0(2)
0i (n)a(n) 0i(n)/?(n) ip2(n)ct(n) ipn/2(n)P(n)
This is often referred to as a Slater determinant.
The only problem now we have is that how to solve those equations: for each
electron the potential due to all other electrons has to be known, but initially
none of these is known. In practice trial orbitals are used which are iteratively
modified until a self-consistent solution(the so-called Self-Consistent Field or
SCF) is obtained, which can be expressed as a solution to the Hartree-Fock
equation:
(Hj + vr)4>i = Fifr = tih
(2.11)
12
The HF equation illustrated above also can be expressed in algebraic Roothaan-
Hall form,
N
E(^-e^Ki = 0 (2.12)
v=\
the e,- is the one-electron energy of molecular orbital ipi, is the overlap matrix,
c^i is orbital expansion coefficient, and is Fock matrix.
Molecular Orbitals Theory: the LCAO/MO Method
Molecular orbital theory is an approach to molecular quantum mechanics which
use one-electron functions or orbitals to approximate the full wavefunction. In
almost all practical calculations the orbital are Molecular Orbitals, written as the
linear combinations of basis orbitals(LCAO).
Based on LCAO, molecular Orbital ij> is a linear combination of N basis func-
tions:
N
i>i = (2.13)
/i=i
Note that the expansion of the wavefunction in terms of basis function leads to
a limitation of the accuracy of the ab initio HF theory because there is a limited
number of basis functions available. The greater the number of basis functions
the better the wavefunction, the lower the energy. The limit of an infinite basis
set is known as the Hartree-Fock limit. This energy is still greater than the exact
13
energy that follows from the Hamiltonian because of the independent particle
approximation.
Basis Functions
As we just mentioned above, molecular orbitals need to be expressed as linear
combinations of a finite set of N prescribed one-electron functions known as basis
functions. This helps to reduce the problem of finding orbitals from the complete
descriptions of the three-dimensional function & to only a finite set of linear
coefficients for each orbital. In simple qualitative versions of molecular orbital
theory, atomic orbitals are used as basis functions, which is familiar by the name:
linear combination of atomic orbital theory(LCAO). Two types of atomic basis
functions have received widespread use: Slater-type atomic orbitals (STO), in
which the radial part of orbitals is an exponentially decaying function. The
disadvantage of it is that evaluation of integrals involving such functions is time-
consuming. The second consists of Gaussian-type atomic function(GTO), which
introduced by Boys[15]. They are less satisfactory than STOs but they have
the important advantage that all integrals in the computations can be evaluated
explicitly without recourse to numerical integration. The third possibility is to
use linear combinations of Gaussian functions as basis functions, which is usually
called Contracted Gaussian.
A minimal basis set has one basis orbital per two inner shell electrons and
14
one basis orbital for each valence atomic orbital. For example, the second-row
elements have basis functions like: Is, 2s, 2px, 2py, 2pz atomic orbitals. The STO
functions are represented approximately by a linear combination of n GTOs(STO-
nG) so the limits of the original STO calculation are eliminated. The common
minimal basis set is the ST0-3G basis, which consist of expansions of Slater-type
atomic orbitals in terms of 3 gaussian functions.
In order to increase the flexibility of the SCF wavefunction and to obtain
better results the number of basis functions per atom has been increased. One
example is the double Zeta basis set. These have two functions for each AO
of the minimal basis, one closer to the nucleus, the other allowing for elec-
tron density to be further from the nucleus. For second row elements this
gives Is, Is', 2s, 2s', 2px, 2px', 2py, 2py', 2pz, 2pz' where the prime representing the
adding basis sets. If this doubling of the minimal basis is done only for the va-
lence orbitals we get the so-called split valence basis sets. For second row elements
this uses Is, 2s, 2s', 2px, 2px', 2py, 2py1,2pz, 2pz'. Examples of split-valence basis
functions include 3-21G and 6-31G.
In the 3-21G basis, the Is AO of a second-row element is represented by a
fixed combination of 3 GTOs, the 2s are approximated by a fixed combination
of 2 GTOs and the extra valence orbitals 2s' are a different GTO.
6-31G is just bigger, as for second row elements again, it has 6 GTO functions
for Is; 3 GTOs for 2s orbital; and 1 GTO function for the 2s' orbital. They can
15
give a much lower energy, but at the expense of having more integrals to be
calculated. The 6-31G set became the most often used starting basis set for ab
initio work sd far for its reasonable compromise between computational burden
S.
and quality of results.
The next step after 6-31G usually is to go to basis sets which have polarization
functions, eg. d-orbitals for second-row elements, which can allow for a lower
symmetry of the electron distribution in a molecule compared to an atom. The
6-31G* (6-31G plus d-functions for second-row atoms) basis sets are at moment
considered to give good quality geometries for most molecules.
Hartree-Fock Theory
We just discussed how to construct a determinantal wavefunction from molecular
orbitals, and how those orbitals expand in terms of a set of basis functions. Up
to this point, we have not explained the Hartree-Fock theory, which based on the
variational method, is employed to help solving the Schrodinger equation. HF
method approximates the true many electron wavefunction as a single determi-
nant of rnormalized-ortho spin orbitals. Its important merit is ease of application,
and its principle deficiency is that it does not always give the lowest possible en-
ergy. According to the Variation Principle, the expectation value of the energy
16
evaluated with an inexact wavefunction is always higher than the exact energy.
< V\H 1$ >
< $1$ >
= E$ > Eycxact
(2.14)
As a consequence the lowest energy is associated with the best approximate
wavefunction and energy minimization is equivalent with wavefunction optimiza-
tion. Eigenvalues E are interpreted as orbital energies. Due to the Koopmans
theorem[16], those have an attractively simple physical interpretation: they give
the amount of energy necessary to take the electron out of the molecular orbital,
which corresponds to the negative of the experimentally observable ionization
potential.
The HF equations can be rewritten as follows:
CiyFpv = Â£i CivS^v (2.15)
V V
in which the overlap matrix N x N is
sr Wvlik') = (/I")
(2.16)
and the N x N Fock martrix is:
Fv = (fi\F\i/) = (n\Hf\v) + '52P\',[(fiv|A
A, <7 Z
and the P is the density matrix:
P\c 2 y; Cj\Cj0
(2.18)
17
The significance of the density matrix is that it describes the electron density
of the molecule. Note that the criterion for judging convergence of the SFC
procedure can refer to the density as well as to the energy, because both have to
be stationary at self-consistency.
The Fock matrix and the orbitals are not linear in the equation above, because
both depend on the molecular orbital expansion coefficients. Thus, they only can
be solved iteratively. That is why it is named Self-Consistent Field method
because the resulting molecular are actually derived from their own effective
potential.
The ground state energies of Slater determinants(eq.2.10) from a Hartree-Fock
calculation are readily expressed in one- and two-electron integrals:
occ occ
e = 'Lh'h + EK*lw) WIj'O] (2-i9)
i i
where H'h, a matrix which represent the energy of a single electron in a field of
bare neclei, is equal to
H'ii = (il-ff1;) = / (2-20)
and the two-electron integra^uljj) which describes the repulsion between
two electrons each localized in one orbital is called a Coulomb integral, (ij\ij) is
termed as Exchange integral. Their quanties (ij\kl) can be expressed as
(ij\kl) = [ [ ${x\)){x\)k{xi)*i(x2)dxxdx2 (2.21)*
J J ri2
18
In many cases it is advantageous to apply the restriction that electrons with
opposite spin pairwise occupy the same space. This leads to the Restricted
Hartree Fock' method(RHF), as opposed to the Unrestricted version(UHF). For
N..
RHF theory, a single set of molecular orbitals is used, some being doubly occu-
pied and some being singly occupied with an electron of a spin. For example,
a five-electron doublet state, the spin orbitals used in single determinant are
(V,i<*)(V,i/?)(^2<*)(V,2/#)(V3q:)- This is illustrated in Figure 2.1 (a). However, de-
tails are more complicated since different conditions apply to singly- and doubly-
occupied orbitals. Hence another theory are brought out which in common using
for open-shell systems: spin-unrestricted Hartree-Fock theory The example we
illustrated above now change to such a diagram(Figure 2.1 b):
As we can see, there are two distinct sets of molecular orbitals tpf and Vf-
The electron configuration for same five-electron doublet may be written as:
/?)('0Â£o:)(^>2/?)(V,3a)- Its important to note that the previously doubly-
occupied orbital ipi now is replaced by two distinct orbitals, and making
it a more flexible function, thus can approximate the exact solution better. Since
the RHF function is a special case of the UHF function, it follows from the vari-
ational principle that the optimized UHF energy must be below the optimized
RHF value. However, for UHF wavefunction, sometimes they are not true eigen-
functions of the total spin operator, unlike exact wavefunctions which necessarily
are. On the other hand, it turns out to be an important advantage for RHF
19
'04 b r<|
''"-03 H- 03 03
Increasing Increasing
Energy , Energy
02 02 02
0i | b 01 0f
(a) (b)
Figure 2.1: Electron configuration, diagram for (a) (V>ia:)(0i/^)(02a)(02/?)(03QO
and (b)
method that magnetic moments associated with the electron spin cancel exactly
for the pair of electrons in the same spatial orbital, so that the SCF wavefunction
is an eigenfunction of the spin operators. In practice RHF is mostly used for
closed shell system, UHF for open shell species.
The total energy for a closed shell ground state RHF model can be written
as:
occ occ occ
E = En + 2 Â£ Hu + H't'b'j) EWW) (2-22)
i,i i,i
20
Population Analysis
Atomic charge is kind a concept widely used in molecular mechanics, however
it is really hard to use a unique definition of atomic charge in a molecule. All
ways are to a certain extent arbitrary. We will start at a way called Mulliken
Population Analysis.
The electron density distribution is:
p(r) = Z Z (2.23)
I1 V
Integration of this over entire space lead to :
N N
Ur)dr = Y.Y,P^ = n (2.24)
* fl V
Here the is the overlap matrix over basis functions. The total electron count n
is then composed of individual terms P^S^. The integration of equation (2.23)
can be separated into diagonal and off-diagonal terms:
N N N
E^+2EE^- = " (2.25)
II II __
where the former part, represent the net population of the basis orbitals and__
the latter part, are the overlap population Q^:
Qnv = 27^ v) (2.26)
For Mulliken scheme, the overlap population is just shared between the con-
21
tributing atoms, which leads to the following charges for each basis orbital.
= Pw + E (2.27)
This is just gross population for <^, the sum of gross population of all N basis
function is equal to the total atomic charge.
Mulliken population analysis has an important disadvantage: the extended
basis sets can be arbitrary, leading to unphysical results, e.g. charges of more
than 2e, which result from the fact that the basis orbitals at center of one atom
actually describe electron density close to another nucleus. That is why we choose
Baders Atoms In Molecules(AIM) method as the algorithm for our atomic charge
distribution.
The topological theory of atom in molecules offers a concise description of
the electronic structure of molecules in terms of the topological properties of the
electron density. Based on such a theory, atomic basins are defined as the regions
in Cartesian space that are bordered by zero-flux surfaces in the gradient of the
electron density. Such partitioning allows one to treat the resulting atoms as
quantum mechanically separate systems. The properties of atoms in molecules
are calculated by integration over the atomic basins[17]. In particular, for an
atom A, the corresponding atomic bcLsin(fl>i), the two spin orbitals (i and
the corresponding element of the atomic overlap matrix(AOM) is given by
22
(2.28)
(*l j) = / i(r)Mr)dr
JaA
When the spin orbitals are related to the Hartree-Fock occupied spin orbitals
through a unitary transformation, the elements of AOM satisfy two important
conditions:
EW = (*'b) = Sij (2-29)
and
Na = EW)-*
i
(2.30)
N = J2na
(2.31)
A
where Na is the number of electrons of the atom A and N is the total number
of electrons present in the system under consideration. The difference between
Na and the atomic number of A is known as the Bader atomic charge.
Moller-Perturbation Theory
Restricted Hartree-Fock SCF theory has some painful shortcomings. Consider
for example the dissociation of the H2 molecule:
Hf + H~ i H H * H. + H.
A dissociation failure occurs because the separated hydrogen atoms cannot be
described using doubly occupied orbitals, so that H2 tends to dissociate in H+
23
and H~, which can be described with a doubly occupied orbital on H~. Without
occurring in the UHF theory, it becomes a big problem that it does not give pure
spin state in RHF.
s..
Another limitation of the HF method in general is that due to the use of the
independent particle approximation that instantaneous correlation of electrons
is neglected, even in the Hartree-Fock limit. It has been recognized that bond
dissociation energies are seriously underestimated if this correlation is not ad-
equately taken into account, sometimes even little difference maybe chemically
important.
Several approaches are known that try to calculate the correlation energy
after HF calculations, such as Configuration Interaction(CI), Moller-Plesset Per-
turbation Theory(MP) and Multi-Configuration SCF(MCSCF). Here we will just
focus on the Moller-Plesset theory which is very popular in recent years.
The perturbation theory of Moller and Plesset, closely related to many-body
perturbation theory, is an alternative approach to the correlation problem. Within
a given basis set, its aim is to find the lowest eigenvalue and corresponding eigen-
vector of the full Hamiltonian matrix. The basic idea is that the difference
between the Fock operator and the exact Hamiltonian can be considered as a
perturbation:
H = F + V
24
Corrections can be made to any order of the energy and the wavefunction:
E Ehf + Ei + E2 + E3 4- (2.32)
s. = $ tfF + ^1 + + $3 + ' (2.33)
This probably represents the simplest approximate expression for the corre-
lation energy. Its enormous practical advantage is that FAST, while it is rather
reliable in its behavior, and size consistent. A disadvantage is that it is not varia-
tional, so the estimate of the correlation energy can be too large. In practice MP2
must be used with a reasonable basis set such as 6-31G* or better. Subsequent
MP-levels MP3, MP4(usually MP4SDQ) are more complicated and much more
time-consuming.
Quality of ab initio Results
The philosophy of a model chemistry is that it should be uniformly applicable
and tested on as many systems as possible. Some requirements are necessary to
be satisfied. They could be summarized as:
1. well defined and applicable in a continuous manner to any arrangement of
nuclei and any number of electrons.
2. no matter what kind method of configuration selection is employed, it must
not lead to such a rapid increase in required computation with molecular
size as to preclude its use in systems of chemical interest.
25
3. model need to be size-consistency.
4. the calculated electronic energy be variational.
We should realize that practical models incorporating electron correlation do
not usually satisfy all the requirements. In the rest of this part we will focus on
the performance of model chemistries that can be practically applied for organic
molecules with nowadays hardware and software, that is HF and MP2 methods
with basis sets usually limited to the 6-31G* level.
Geometry
As far as equilibrium geometry is concerned, HF and MP2 ab initio models can
reach to excellent results even with the modest basis sets. HF/6-31G* or MP/6-
31G* are good enough to be considered reliable for the geometries of organic
molecules. The bond length calculated at the HF level is usually overestimated
by ca.0.01 0.02 A as a result of the neglect of electron correlation. This was
discussed more detailed in reference [18]. Numerical errors in calculating geome-
tries arise from rounding, insufficient accuracy in integral evaluation, incomplete
convergence of the SCF procedures and incomplete convergence of geometry op-
timization. Efforts such as carrying out the computations at higher precision are
limited because there is little point in proceeding too far beyond the limits of
accuracy imposed by the systematic errors of the methods used.
26
Vibrational Frequencies
The exploration of chemical reaction pathways provides a great challenge for
theory. This-involves a characterization of structures and relative energies not
only of stable forms, but also of other stationary points on the potential surface,
corresponding to reactive intermediates and transition structures. Calculated
normal-mode vibrational frequencies play important roles in the use of theory as
a means of characterizing molecular potential surfaces. Due to the availability
of analytic second derivatives of HF and MP2 wavefunctions, the calculation has
become almost a routine matter. It turns out that the results even with modest-
basis HF models are quite good. The frequencies are consistently overestimated,
error source including inherent inaccuracies of differentiation techniques required
in evaluation of the matrix of force constants, and from uncertainties in the
selection of equilibrium geometry. Uniform scaling of the computed frequencies
by a factor of 0.89 0.01 gives a good agreement for most cases. For MP2 the
scaling factor should be closer to 1.0.
Energies
The accurate computation of absolute or relative energies remains a major chal-
lenge. Even conformational energy differences and barriers are not reliably com-
puted with HF or MP2 models using small basis sets (6-31G* or smaller). The
27
error could arise from the adoption of equilibrium geometry. While it is certainly
desirable for a theory to give a bond dissociation energies, failure to do so does
not preclude its use for the prediction of reaction thermochemistry. Energies
of reaction can be predicted relatively accurately, especially for isodesmic pro-
cess. Of course the demands on accuracy are very high, on the other hand, a
comparison between related systems can often be made quite well.
Atomic Charge
Comparing to the other method, Baders AIM method is one of the most recom-
monded choice in representing the polarization of bonds. However, a defiency of
this method is its dependence on atomic size that lead its populations exaggerate
electron densities at electronegative atoms. It is really hard to say which one is
realistic in certain, however, numerous attempts have been made to obtain better
descriptions of atomic charges.
28
CHAPTER 3
COMPUTATIONAL METHODS
In this chapter, the computational methods used in this work are described.
The Schematic Route
We just discussed the theory about the calculation, now we are facing the chal-
lenge that how could translate the mathematical formalism to a step-by-step
instruction on a digital computer. Some requirement has to be satisfied for this
kind algorithm or program, such as the efficiency, calculation time, convenience,
and flexibility. Almost all chemical programs are written in FORTRAN, a kind
language used frequently for scientific programs. Most widely used program nowa-
days are Gaussian, Spartan, HONDO, IBMOL, etc. They are quite different in
many fundamental respects, ranging from overall design to the algorithms. Here
we try to use a scheme(Fig.3.1) to explain the logical structure of a typical pro-
gram of executing the types of molecular orbital calculation described in the
theory section. Our goal is to present a fundamental path to the efficient appli-
cation of available programs to the solution of chemical problem.
In the schematic representation, circle and rectangles represent two different
29
logic elements. The circle, namely mass storage, consist of several different logic
units, one of random-access type for the storage of the various matrix that appear
during the calculation. Others including some greater capacity and sequential-
V.
access type for the storage of two-electrons integrals. The latter is called program
elements. Each independent element perform a specific job. They interconnect
with each other by a master element named control program, and exchange data
via mass storage. The advantage of this feature enable the development and
changes of program without affecting other sections.
The calculation starts by specifying the molecular geometry, followed by se-
lection of basis set. If it is standard, it is pre-stored in computer, if not, it must be
specified in detail for each atoms. At this point, the one-electron overlap, kinetic
energy, potential energy integrals, and the two-electron repulsion integrals, all of
which will be required for the solution of the Roothaan-Hall equation(Eq.2.12)
are calculated. As a starting point for the solution of the self-consistent-field
equations, a guess at the wavefunction or at the density matrix(Eq.2.18), is re-
quired. A solution to the Roothaan-Hall equation may now be attempted. Since
these equations are not directly soluble, an iterative procedure must be used, the
time required for their solution is directly proportional to the number of itera-
tions taken. Once self-consistency has been achieved, the wavefunction may be
printed, and a population analysis carried out. Finally, the Hartree-Fock study
30
Figure 3.1: Logical Structure of ab initio molecular orbital program
31
may be followed by a calculation of the correlation energy.
Geometries
We just discussed the overall landscape for the program execution. Now we will
go a little detailed and try to provide a brief description of the method which we
really used throughout our calculation.
The geometry read usually in two ways: one is directly putting the data
as Cartesian coordinate; another way is in the form of Z-matrix, a method of
defining a molecule atom by atomic numbers, bond lengths, bond angles, and
dihedral angles. One thing should be noticed here is that the Z-matrix in no
way defines the bonds to be formed in the calculation, it is simply a geometrical
device used to define the positions and types of the atoms. In our calculation, the
molecules were first created through a program called SPARTAN, and optimized
at 6-31G** level, then the Cartesian coordinate were taken to the Gaussian94
which is stored in DIGITAL. Some molecules were input by Z-matrix, and the
parameters such as bond length, angle and dihedral angle were also brought from
the calculation in SPARTAN. As a starting point, all geometries were optimized at
Gaussian94 at MP2 level, and were further probed by calculating the frequencies
in order to ensure that these structures were minimal.
32
Basis Set
Our calculation were started at MP2 level with the standard 6-31G* basis set
storing in Gaussian 94 software. After initial geometry optimized, the single point
calculation of frequencies and atomic charge followed by, later one is at a higher
level 6-311G**. The donation for such a base set is: MP2/6-31lG**//MP2/6-
31G*, meaning a single point calculation with the MP2/6-311G** basis, at a
geometry optimized with the MP2/6-31G* basis set. The basis sets before double
slash represented the final single point calculation and after double slash is for
the initial geometry optimization.
The basis sets of most atoms involved in our calculation have been installed
in program, the calculation just need to search the right level basis automati-
cally and bring them into computation. However, this is a little complicated for
bromine, since there is no basis set for it at MP2/6-31G* and higher level. The
calculation then need to be set manually. Instead of searching the basis sets from
the program, the computer takes the sets in command file, where they have been
specified for each atoms.
Energy
The energies of all molecules were determined using second order Moller-Perturbation
theory(MP2) optimized geometries with single point higher level basis set. The
33
preliminary calculation help save lots of time needed for the higher level compu-
tation. Though its not the goal for our analysis, it has to be carried out at first
step calculation, where all other parameters can only be obtained after it is done.
v._
Frequency Calculation
The reason we put frequency into calculation is to ensure a zero-point structure,
or a minimum energy achieved. This is essential because only the structure with
no negative frequency can be considered as a groundstate molecule, other wise
one only can take the structure as the transient state(one negative frequency) or
excited state(> 2 negative frequencies). This could be done easily by inputing a
command for frequency calculation and the program will execute automatically.
Atomic Charge
Atomic charge are calculated by the integration. We adopted the Baders Atoms
In Molecules[8], which need a input file named as: file.wfn(wave function). This
could be done by assigning a output=wfn option in gaussian file. As an initial
reading file, wfn file will be taken to help select critical points for the molecules,
then followed by executing a proaim(properties of atoms in molecules) program.
In order to make sure the result is reliable, one should check the L value listed
in the output file if they are all smaller than 0.003. Also, the total number of
electrons integrated should be the expected number of electrons for the molecule,
34
usually the sum is within 0.01 of an electrons of the expected sum. Both of these
two parameters, alone with atomic charge will be tabulated into an output file
as the final result.
35
CHAPTER 4
RESULT AND DISCUSSION
Our calculations are summarized in the following three parts:
1. AA3(including C4 ), where A=B, C, N, and P\ X=H, F, Cl and Br .
2. AY3 (including CY4 ), Y=OH, SH, NH2, and PH2.
3. POX3, X=H, F, Cl, Br.
The energy of both optimized structures at the MP2/6-31G* level and single point
calculations at the MP2/6-311G** level are listed in Table 4.1. No imaginary
frequencies have been found in any of our computations; hence, we can conclude
that all molecule are stable ground state molecules.
The energies listed in Table 4.1 are consistent in that the single point ener-
gies(without changing geometry) at a higher level are lower more stable molecules
than the lower level basis set calculations. The following discussion addresses each
class of compound studied.
36
Table 4.1: The optimized energy and single point energy(in au)
molecule-- a b molecule a b
bh3 -26.46424 -26.49578 B(OH)3 -251.78973 -251.96787
bf3 -323.77866 -323.98933 B{SH)3 -1219.49120 -1219.63371
bci3 -1403.72408 -1403.83850 B{NH2)3 -192.20144 -192.34756
BBr3 -7742.43392 -7742.80667 B(PH2)3 -1050.71141 -1050.86992
CHa -40.33255 -40.38053 C{OH)a -340.49044 -340.72770
CFa -436.44755 -436.73095 C(SH)a -1630.81665 -1631.01437
ecu -1876.40717 -1876.56603 C{NH2)a -261.08775 -261.28313
CBrA -10317.37904 -10327.97209 C(PH2)a -1405.90155 -1406.12086
nh3 -56.35421 -56.41092 N(OH)3 -281.30316 -281.49322
nf3 -353.22678 -353.45022 N(SH) 3 -1249.17177 -1249.32784
nci3 -1433.30442 -1433.42613 N(NH2)3 -221.82464 -221.98284
NBr3 -7772.06093 -7772.44021 N(PH2)3 -1080.54438 -1080.71890
ph3 -342.55171 -342.61274 P(OH)3 -567.74731 -567.93226
pf3 -639.75168 -639.96216 P(SH) 3 -1535.48269 -1535.63373
PCl3 -1719.70387 -1719.82665 P{NH2) 3 -508.17539 -508.33911
PBr3 -8050.90004 -8058.39636 P(PH2) 3 -1366.76985 -1366.93712
poh3 -417.59563 -417.69594 pof3 -714.83321 -715.08231
POCh -1794.77443 -1794.93467 POBr3 -8133.47158 -8133.49714
a. calculations were carried at MP2/6-31G*//MP2/6-31G*.
b. calculations were carried at MP2/6-311G**//MP2/6-31G*.
37
Trihalide and Hydrogen Compounds AX3
The results of this section will be presented separately for different As. X
includes theLbree 7r donor halogens(F, Cl, Br). Hydrogen, though not having a
p electrons for the back donation, has been calculated as a comparison of effects
due to electronegativity.
BX3
The valence electronic structure of boron is 2s22p. However, instead of monova-
lent, boron is always trivalent, because forming a three bonds compound BX3
can release more energy than a single bond BX, even some energy have to be
used first promoting one s electron to a hybridized valence state of the sp2 type,
wherein the three sp2 hybrid orbitals lie in one plane at angles of 120 [3]. It
would therefore be expected, and is indeed found, without exception, that all
monomeric, three-covalent boron compounds are planar with X-B-X bond angles
of 120.
The Lewis acidity of boron trihalide arises from the empty 2p orbitals on
boron. Its trend varies as: BF3 < BCI3 < BBr3, which has been explained by
back-donation. If this is true, we should expect the charge on boron to be less
positive for BF3 than BCI3, since back pir pir bonding from fluorine to boron
will neutralize the positive charge caused by a binding with F.
38
Table 4.2: The bond length(B-X), bond angle(X-B-X), atomic charges(Boron)
and charge capacities of BX3.
molecule bh3 bf3 BCl3 BBr3
distance^ ) 1.192 1.324 1.737 1.900
atomic charge(jB) 2.00 0.60 0.13 1.74
charge cap. [5]( eV~x ) - < 0.063 0.089 0.103
bond angle() 120 120 120 120
The B-X bond length and angle(X-B-X) of the optimized geometries as well
as the atomic charges on boron are listed in Table 4.2. These structures and their
charge distribution are detailed in Figure 4.1.
In Table 4.2, we see that the bond lengths increase from hydrogen to bromine
which is due to the expanding atomic size of the X group. The calculations show
that the atomic charge on boron varies as: BF3 > BCl3 < BBr3, indicating that
boron is significantly more positive in BF3 than in BCl3, as we can see in Figure
4.1. This result is consistent with a study by Brinck et al., who carried out
MP2/6-31+G**//MP2/6-31+G** level calculations, where the atomic charge
on boron is 1.491 in BF3 and 0.291 in BCl3[5]. Obviously the back-donation
approach cannot explain these results. Instead, the term charge capacity k
has been used to explain the anomalous acidity trend of boron trihalides. Here,
39
Figure 4.1: The optimized structures and atomic charges of BX3
40
the charge capacity, k, is derived in terms of the ionization potential, /, and
electron affinity, A, which are measures of an atom or groups ability to accept or
donate electronic charge. Consideration of charge capacities has made it possible
'v__
to explain a number of seemingly anomalous aspects of chemical behavior. Their
calculations of charge capacities are listed in Table 4.2. As we can see, the value
of k varies in the same direction as the acidity trend, which is also the same as the
trend of electron affinity increasing from F to Br. Thus the trends in the latter
can be understood in terms of the increasing ability of the larger, more polarizable
halogens to accommodate additional electron density. This is not unusual since
as pointed out by Brinck[5] the electron affinities increasing for heavier congeners
within the same column of the periodic table. At this point, lets go back to the
definition of electron affinity: the negative of the energy change accompanying
electron gain. Hence, a negative electron affinity corresponding a fact that hard
to gain electrons. Indeed, most small closed-shell molecules containing mainly
first-row atoms, like BF3, have negative electron affinities. The reason Brinck
and his coworkers prefers the charge-capacity to the back-donation approach is
because it is consistent with the Lewis acidity trend of boron trihalides.
A very striking number in Table 4.2 is the extraordinary high atomic charge on
B in BBr3( 1.74). This appear to indicate that the bromine atom draws electron
density to it even stronger than fluorine does. Since this is the first time the
AIM method for BBr3 has been carried out, we cannot compare any previous
41
work. Unfortunately, the result is questionable because it is not consistent with
expectation. It has been stated that the overlap between pn orbitals increase
as 2px 3pit> 2pit 4p7r[4]. This increasing % overlap effect would leave a
less positive charge on B in BBr3. In addition, the fact that bromine is less
electronegative than chlorine should result in a less positive charge on B of BBr3
too. As we mentioned in introduction, the AIM charges are too large for the
spherically symmetrical atoms if they represented point charges. A qualitative
explanation for such large charge is that the electron density near a small more
electronegative atom is highly concentrated; on the other hand, a greater volume
is assigned to a larger and less electronegative atom and, as a result, the Bader
population on that atom is exaggerated[10]. The greater Bader population arises
not only because of electronegativity differences but also because the difference
of atomic sizes, which ought to be irrelevant. Even so, we cannot confirm this
as the explanation for the abnormal large atomic charge on boron and bromine
until more detailed work can be carried out.
CX4
Since carbon in CX4 has no vacant p orbitals, it represents a different system for
examining the effect of substitents on charge distribution.
All halides of carbon have a tetrahedral structure. Since there is no empty
orbital on carbon to accept any tt back donation, one will expect that any change
42
effect could only come from the different electronegativities between C and Xs.
Although no CX4 charge studies have previously been carried out, researchers
have calculated the atomic charges for CXÂ£, a different orbital system from CX4,
but one that is isoelectronic with BX3. Their NBO method analysis show that
the charge of C varies reasonably with the halogen electronegavities: CFÂ£ >
CClt > CBr+[ 11].
As we can see from the Table 4.3 and Figure 4.2, the structures are all tetra-
hedral for the four compounds we have studied. A little difference among the
angle of both CCI4 and CBr4, but they are all very close to the tetrahedrals.
Charge distribution is consistent with the trend of electronegativity of X. For
CH4, it is in agreement with the experimentally determined results with a bond
dipole in the sense C+ H~. For carbon halides, the most electronegative atom
is fluorine, which has a -0.70 charge and gives the central carbon a 2.78 charge
while the atomic charge of carbon in CCI4 is only 0.38. This is in agreement with
their electronegativities(F: 3.94, Cl: 3.16) [19].
The negative charge on carbon in CBr4, seems to indicate that carbon is more
electronegative than bromine! The negative charge on carbon is understandable
for the CBrt[ 11], which is a net result of a + ir effects. However, unlike CBrt,
which has an empty orbital on the carbocation, the carbon in CBr4 could not
accept any back donation from bromine. Thus only a bonding can occur and
only a positive charge on carbon should be expected. It is still questionable
43
b
d
Figure 4.2: The atomic charge of CX4
44
Table 4.3: The bond length(C-X), bond angle(X-C-X), atomic charge(Carbon)
and Charge Capacities of CX4
molecule CHA cfa ecu CBta
distance(A ) 1.090 1.331 1.771 1.900
atomic charge(C7) 0.12 2.50 0.38 -0.02
bond angle() 109.47 109.47 109.47* 109.47*
* average value.
that this negative charge is a result of the deficiency of the AIM method. More
investigation is needed to give us a reasonable explanation.
NX.3
Unlike electron deficient boron and electron precise carbon, nitrogen has a lone
pair of electrons. Based on Gillespiess electron repulsion ideas, we would ex-
pect these compounds to be pyramidal. Like CX4, there should not be any 7r
back donation, because, nitrogen has no empty orbitals to accept 7T electrons
from X. Hence, the charge distribution is expected to result from the different
electronegativities of N and X.
Compared to the electronegative fluorine, hydrogen can donate its electrons to
nitrogen(Afif+) when they bond together. In the ammonia molecule, the neg-
45
Table 4.4: The bond length(N-X), bond angle(X-N-X), atomic charge(nitrogen)
and charge capacities For NX3
molecule NH3 NFZ NCh NBr3
distance(A) 1.017 1.385 1.775 1.885
atomic charge(iV) -1.01 0.53 -0.27 -0.46
bond angle() 106.32 101.67 107.58 111.97
ative charge which arises from the inductive effects of the hydrogen atom, make
the lone pair electron of nitrogen more available for the donation, resulting in a
Lewis basic NH3[2Q], All nitrogen trihalides are Lewis bases too; however, NF3
is a very weak one compares to NCI3 and NBr3. The strong electronegativity
and the small size of fluorine have a significant effect on the nitrogen, making the
lone pair of electrons on nitrogen much less available. Cl and Br, on the other
hand, are both less electronegative, making NCI3 and NBrs relatively stronger
Lewis bases.
Our calculation can be found in Table 4.4 and Figure 4.3, where all the struc-
ture are close to tetrahedral, and the charge distributions are reasonable with
respect to the different electronegativities of X.
The atomic charge of nitrogen in ammonia is -1.01, which is in agreement
with its Lewis basicity. The charges for nitrogen atom of the three trihalides
46
b
d
Figure 4.3: The optimized structures and atomic charge for NX3
47
are 0.53 for NF$, -0.27 for NCI3 and -0.63 for NBr3. We expect that the in-
ductive effect will decrease with the decreasing electronegativity from fluorine to
bromine, hence, the atomic charge on N will also decrease, which is reflected by
s.
the calculation. Since there is no back donation, the negative charge of nitro-
gen of NCI3 and NBt3 suggests that the nitrogen is more electronegative than
chlorine and bromine when it binds with these two atoms. It is understandable
for NBt3 because nitrogen is indeed more electronegative than bromine(jV:3.04
and Z?r:2.96)[19], but for NCI3 it is difficult to explain. The large size of chlorine
could be a factor, however.
PXZ
There is a big difference between molecules of the NX3 and PX3 types because,
unlike nitrogen, the phosphorous has vacant 3d orbitals of low energy, making it
not only an electron donor, but also an electron acceptor. Thus, PX3 compounds
usually are considered as 7r-bonding ligands [3], and act as mild Lewis acids
toward bases such as trialkylamines and nitro halides. For example, phosphorous
trifluoride, PF3, which sometimes resembles the CO molecule, is a weak a donor
but a strong ir acceptor. This character is attributed to a P-F antibonding
LUMO, which has mainly phosphorous p-orbital character when P is bonded to
the electronegative F atom [19]. As a result, we should expect that atomic charge
on phosphorous in PX3 would be quite different from its analog of nitrogen.
48
Figure 4.4: The optimized structures and atomic charge for PX3
49
Table 4.5: The bond length(P-X), bond angle(X-P-X), atomic charge(P) of PX3
molecule ph3 pf3 PCl3 PBt3
distance(A) 1.415 1.595 2.055 2.251
atomic charge(P) 1.70 2.30 1.42 0.85
bond angle () 94.64 97.63 101.08 101.88
The result of our calculations are listed in Table 4.5 and Figure 4.4. Similar to
the geometry of NX3, all PX3 compounds adopt pyramidal structures. Different
from NX3, however, the atomic charge on P of all three trihalides are highly
positive, in agreement with their Lewis acidity, which we mentioned above. The
varying trend of atomic charge on P is consist to the electronegativity of halogen,
where it decreases as PF3 > PCl3 > PBr3.
The different basicity between NH3 and PH3 is in harmony with the idea
that the N-H bonds and lone pair of NH3 can be regarded as approximately
sp3 -hybrid orbitals, whereas the lone pair in PH3 appears to have much more
s-orbital character and the P-Hs bonds correspondingly more p-orbital character.
50
Table 4.6: The atomic charges of A in AY3.
molecules -OH -nh2 -SH -ph2
by3 2.35 2.28 1.67 0.38
cy4 2.15 1.34 -0.36 -
ny3 0.41 -0.03 -1.25 -2.01
py3 2.19 2.06 0.7 -0.05
AY3, Y = OH, SH, NH2, PH2
Compared to the study of AX3 and AX4, the 7r-donor substituent Y (Y=OH,
SH, NH2, PH2)in AD3 and AD4 have rarely been reported. Most studies only
focus on the structures of the compounds[21] [22] [12]. Since little has been
mentioned about atomic charges, we just present our own calculation and their
quality can only be examined in future studies.
Our calculations of atomic charge of A in AY3 and AY4 can be found in Table
4.6, and detailed charge distribution along with optimized structures are shown
in Figure 4.5-4.8. These results will be discussed individually.
51
BYZ
BYz would be expected to have a coplanar geometry for all the non-hydrogen
atoms, which-.come from the same reason of BX3. However, in a previous work
reported by Volatron and Cemachy[21], B(NH2)3 and B(PH2)3 both have a
slightly out-of-plane structures. According to their explanation, this results from
the competition between conjugation of the lone pairs with 2p vacant boron
orbital and the lone pair repulsion between the three doubly occupied orbitals on
N and P.
The atomic charge on boron are expected to follow the trend of electronega-
tivity of K, although their charge distribution should be a combination of both
electron-withdrawing effect and 7r back-donation effect from Y.
In Figure 4.5 we see that our calculations are consistent with the previous
work, where B(OH)3 and B(SH)z have a planar structure(exact 120 for X-
b-X) while B(NH2)3 and B(PH2)3 have a out-of-planar structures. The twist
angle is about 16.8, close to the Volatrons work(13.1). The atomic charge
on boron is decreased as B(OH)3 > B(NH2)3 > B(SH)3 > B(PH2)3 As
reported by Schmidt et al.[23], the ability of forming strong w bonds is in the
order 0 > N S > P. The close atomic charge on boron of B(OH)3 and
B(NH2)z are 2.35 and 2.28 respectively, in agreement with these two atoms
electronegativities(3.44 for 0 and 3.04 for A^) [19]. The fact suggests that the
52
b
Figure 4.5: The optimized structures and atomic charges of BY3. a: YNH2, b:
Y=PH2, c: Y=OH, d: Y=SH.
53
7r bonding is less effective than the a bonding, at least, not as strong as their
analogs, where the 7r back donation from P(B(PH2)3) and S(B(SH)3) lower the
charge on boron. Otherwise, the atomic charge should be less positive on B of
S.
B(OH)3 due to the negative 7r donation. Kapp and coworkers, whose study focus
on the + CH2XH(X = 0,S,Se,Te) had come to the same conclusion in their
paper[24].
CY4
There is no previous study of the atomic charge of CD4 compounds. Our AIM
calculations are the first studies of this type molecules. Even though we have
been able to calculate the geometry and energy of C{PH2)4, we have so far been
unsuccessful calculating its atomic charges.
A tetrahedral structure except hydrogens is still expected, and the charge
distribution is also expected to be the only affected by the different electronega-
tivities of Fs.
Comparing the charges of three remaining molecules, our results look in rea-
sonable agreements with related boron compounds, as we can see from Figure
4.6. As the electronegativity decreases O > N > S > P, the charges on carbon
decrease in the same order: C(OH)4 > C(NH2)4 > C(SH)4. Even without the
atomic charge of C(PH2)4, we still can regard that the AIM method as success-
fully reproducing the electronegativity of these compounds.
54
d
Figure 4.6: The optimized structures and atomic charges of CY4. a: YNH2, b:
Y=PH2, c: Y=OH, d: Y=SH.
55
ny3
N(NH2)3 and N(PH2)3 are still unknown compounds. One of the big differences
between HY^and NY3 can be seen from there optimized structures: for BY3, they
all keep an almost planar structures for all non-hydrogen atoms. However, for
NY3, the lone pair electrons from nitrogen repelling each other, just as NX3,
a pyramidal configuration for all of these compounds is expected(Figure 4.7).
No 7r back donation should be expected either, since no empty orbitals exist in
nitrogen. In Figure 4.7, the structures from our calculations are in reasonable
agreement with the expectation.
The atomic charges on nitrogen have the same trend as BY3 and CY4: N(OH)3 >
N(NH2)3 > N(SH)3 > N(PH2)3, consistent with the electronegativities of the
substituent. However, unlike B(NH2)3 and B(PH2)3, where the charge distribu-
tion of -NH2 and -PH2 is equal for all three N{ot P) and the six Hs, they are
slightly different in N(NH2)3 and N{PH2)3, suggesting that structures are not
symmetric constructed.
py3
The structures of P(NH2)3 and P(PH2)3, P(OH)3 and P(SH)3 seem quite differ-
ent, as we can see from Figure 4.8. These results again demonstrate the difference
between phosphorous and nitrogen, even though they are both in the same group.
56
Figure 4.7: The optimized structures and atomic charges of NY3. a: Y=NH2; b:
Y=PH2-, c: Y=OH- d: Y=SH
57
Thus the structure with C3 symmetry is the most stable for N(NH2)3 but not
for phosphorous series. The different between the minimum energy structures of
P(NH2)3 and'its homolog N(NH2)z gives a strong indication that the reason for
the energetic preference for the Ca structure of P{NH2)z can be attributed to the
same type of interaction as causes the difference between the planar geometries of
nitrogen centers bound to second-row elements and the pyramidal ones of nitro-
gen centers bound to first-row elements [25]. Probably the best explanation which
has been found for these effects is that of negative hyperconjugation including
the nitrogen lone pairs of electrons[26]
The atomic charges on P in PY3 vary as: P(OH)3 > P(NH2)3 > P(SH)3 >
P(PH2)3, just the same as the second rows B, C, and N. Thus, all reproduced
the electronegativity, even though they have very different structures. Positive
charge on P suggests they could react as Lewis acids. We expect that possibility
of 7r back donation on Ps empty d orbital from the Y substituents; however,
atomic charge calculations alone are not enough to get such a conclusion and
further investigation must be carried out.
Phosphine Oxides POXs(XH, F, Cl, Br)
There have been extensive and renewed interest in hypervalent phosphorous
compounds, focusing on the nature of the bond in phosphine oxides[28][29][30].
Hartree-Fock calculations involving structural effects, d orbital contributions in
58
2.20
Figure 4.8: The optimized structures and atomic charges of PY3. a: Y=NH2; b:
Y=PH2; c: Y=OH; d: Y=SH
59
Table 4.7: The bond length(P-O), atomic charge(P) of P0X3 at MP2/6-
311G**//MP2/6-31G*
molecule poh3 pof3 POCl3 P0Br3
distance(A ) 1.5 1.459 1.477 1.443
atomic charge(on P) 3.25 3.65 2.96 2.11
back bonding, polarity and Mulliken population analysis, have best described the
nature of the P-0 bonds in such studies.
According to Yang and his coworkers[31], the shorten length between P and
F in POF3 results from the electronegative difference affecting the 7r bonding be-
tween P-O. The fluorine withdraws the electrons to its side, resulting a compact
structure of compounds, hence, strengthen the ir back donation from oxygen.
However, as we can see from the Figure 4.10, the charges on 0 are almost same
in all the compounds studied, suggesting that oxygen is not significantly affected
by electronegativity variations of X. The bond length between P and O shows
difference, but not very big. The shortest bond length happened on POBr3
is a surprise, which hard to be explained. Even so, the charge distribution is
reasonable for all four compounds.
The electron-withdrawing power of oxygen lends an electron deficiency to the
phosphorous atom, so that ?r-bond character between P and X can build up. In
60
c
Figure 4.9: The optimized structures and atomic charges of POX3
61
fact the 7r bonding of PX really has a more pronounced effect. When there is no
7r back donation on P from X, the atomic charge of P and X(in this case, X=H)
are 3.25 and -0.61 respectively. When X changed to F and Cl, which both have
p electrons for the 7T bonding, the atomic charge on phosphorous became 3.65 for
POF3 and 2.96 for POCI3. This is in good agreement with the trend of halogen
electronegativity. In addition, it suggests that a 7r bonding could exists between
P and halogens, other wise, the charge on P in POCI3 should be more positive
than that in POH3, because chlorine is more negative than hydrogen, however
a conclusion could not be made before a careful % electron density carried out.
It also proves that the 7r bonding is increased as F < Cl, as same conclusion
as for AX3, which we had discussed earlier. Surprisely, the atomic charge of P
in POBr3 is followed variance trend of halogens electronegativity, making the
calculation of all POX3 be the best.
One should realize that many effects should be considered in studying the
POX3 bond: the (p d)n effect; polarity of the whole compound; radical :PX3
stability; substituent X effects and hybridization. Our calculation is only about
one properties. All we can conclude here is that the different electronegativity of
substituent X has an effect but not significantly on the bonding strength between
PO and the resulting atomic charge on 0. We also believe that there is a tc back
bonding between the 2p electron and the empty d orbitals of phosphorous, which
is reflected by the different charge distribution from both POH3 and POCI3,
62
however, this only can be confirmed by an analysis of 7r overlap on P.
63
CHAPTER 5
CONCLUSION
The atomic charges of a series of molecules have been calculated using Gaussian94
at MP2/6-31lG**//MP2/6-31G* level. These compounds include the halides of
the second rows B, C, and N as well as third row member P. Some x substituents
of these atoms, in addition to the halides have also been calculated -OH, -SH,
-NH2, -PH2, in order to compare the effect on central atoms As atomic charge
of both electronegativity and x back donation.
The structures of all compounds have been optimized to the most stable state,
which was confirmed by a frequency analysis.
Atomic charge of all AX3 and CX4 have been discussed individually. At
this point, I would like to review all the calculations. As mentioned before, the
quality of an atomic charge calculation can be examined by several criteria, one
of them is electronegativity. The all calculations of bromine show a contradiction
with the electronegativity, and the results are so discrete that we dont think
it could be used for conclusion. The discrepancy could come from the Baders
AIM method, or could from the basis sets which specified manually instead of
searching in computer. Obviously current computer program is hard to give a
64
automic charge
Figure 5.1: Summary of atomic charge at A in AX3 and CX4. a: X=H, 6: X=F,
c: X=Cl.
65
satisfied explanation, with the help of development of technique, we hope this
problem could be solved in future study.
On the other hand, the calculations for all other compounds are successfully.
v..
In Figure 5.1, the atomic charges on A in AH3, AF3 and ACI3 are plotted in a,
b, c respectively, when A is carbon, the molecules will be CX3 instead of CX3.
When A bond with halogen fluorine and chlorine, as we see in Figure 5.1 6 and
c, a negative ir back donation decrease the positive charge on B which is caused
by inductive effect of fluorine and chlorine. A lower atomic charge of B in BF3
and BCI3 resulted. When there is no back donation, the results are. in good
agreements with the electronegativities of As, as it shown in Figure 5.1a, where
the atomic charge of second row atoms decrease as B > C > N, and in Fig 5.1 b
and c, C > N. Phosphorous has a more positive atomic charge in all calculation,
which could be explained by its expanding size.
The results for the 7r substituents system AY3 are looked good too. They all
show a great harmony with respect to the electronegativity of Y, where all atoms
of A have a charge distribution variance as: A(OH)3 > A(NH2)3 > A(SH)3 >
A(PH2)3(AY4 if A is carbon), suggesting that the Lewis acidity would decrease
as the same order. This has been described in Figure 5.2, where we can see that
the effect on A from the combination of effect of electronegativity from 7r-donor
substituents Y seems follow the similar trend, which OH and NH2 are more
electronegative than their congers SH and PH2. Unlike AX"3(and CX4), the 7r
66
A{NH2)3 and A(PH2)3{n).
o>
-*
cr*
back donation has a much less effect on boron, because boron is still positive than
carbon, as it shown in Figure 5.2 a and b. The atomic charge calculation is not
enough to find out whether or how the back donation effect on boron, however,
it only can be answered by a more detailed investigation such as a i electron
density analysis.
The electronegativity of X has an insignificant effect of tt bonding between
P = O in POX4, however, it does cause a slightly change on 0 atomic charge
distribution due to the 7r back donation of X and the electronegativity of X.
Further investigation is necessary for a complete description of molecules, such
as the exact charge difference when there is a t back donation. Also, the absence
of high level basis sets for the big size atoms leading a questionable performance
of ABr3 and the problems are hopefully to be solved in future. Despite some
discrepancy of ABr3, the overall results of our whole series molecules show the
calculation is successfully. We believe that the Baders AIM method is still among
the best choice for small systems.
68
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