Part One
Introduction
1.1 Historical Background
The existence of volatile boranes has been known since 1878^{1} but they remained uncharacterised until Alfred Stock and co-workers studied these compounds between 1912 and 1936.^{2} Stock’s development of high vacuum line techniques necessary to handle these air sensitive materials resulted in the preparation and characterisation of B_{2}H_{6}, B_{4}H_{10}, B_{5}H_{9, }B_{5}H_{11}, B_{6}H_{10} and B_{10}H_{14}. Stock was unable to deduce a single structure correctly.
In 1945 Longuet-Higgins introduced the concept of 3-centre 2-electron bonds.^{3} This idea was developed by Lipscomb.^{4} Lipscomb’s formulation of a consistent description of this class of molecule required both the development of experimental techniques (low-temperature single-crystal X-ray diffraction) and the extension of valence theory to multi-centre bonding. Lipscomb also employed ^{11}B-NMR spectroscopy as a major tool for the investigation of symmetrical and fluxional character in boranes. The contribution of molecular orbital theory to the understanding of the structures of boron hydrides has been profound. Lipscomb and his associates developed extended Hückel and PRDDO methods^{5} as well as practical methods for the calculation of NMR chemical shifts^{6} mainly for theoretical work on boranes.
Beaudet examined in a recent review the experimentally determined structural data for boranes.^{7} His review covered X-ray, microwave and electron diffraction data from the literature and showed how the different experimental techniques resulted in small differences in the accurate determinations of geometry. The work highlighted “unresolved accurate molecular structure problems” and suggested gas-phase structure determinations rather than ab initio structural determinations as a method for solving these problems.
In more recent work by Schleyer and co-workers it has been shown that structures obtained through the use of the ab initio method are at least as accurate as experimental methods.^{8} This work combines ab initio structural determination with IGLO (individual gauge for localised orbitals) chemical shift calculation.^{9} This theoretical method uses the degree of agreement between the experimental NMR spectra and the theoretical prediction to evaluate the accuracy of the ab initio structural model.
1.2 Wade’s Rules
Wade’s rules provide an empirical method of structure prediction.^{10} They were originally proposed for the structural determination of boranes and carbaboranes.
Structure type |
Number of vertices of polyhedron. |
Number of cluster electron pairs |
Examples |
closo |
n |
n+1 |
[B_{n}H_{n}]^{2-} |
nido |
n+1 |
n+2 |
B_{n}H_{n+4} |
arachno |
n+2 |
n+3 |
B_{n}H_{n+6} |
hypho |
n+3 |
n+4 |
B_{n}H_{n+8} |
Table 1.1 Wade’s Rules.
A fifth class of structure is conjuncto. This class is not based on a single polyhedron and can by thought of as involving multiple cluster fragments linked together.
The structures consistent with this set of rules are shown in Fig 1.1.
1.3 Isolobal principle
Groups that are isolobal have similar frontier molecular orbitals in terms of symmetry and electron count. Exchanging groups that are isolobal results in identical structural types. This principle is implicit in Wade's Rules.
Electron closo nido arachno
Pairs
6
7
8
9
10
11
12
13
Fig 1.1. Diagram showing classical Wadian structures.^{11}
The neutral species closo-C_{2}B_{4}H_{6} is isoelectronic with [B_{6}H_{6}]^{2-}. The fragment {BH^{-}} is isolobal with {CH}. The exchange of two {BH^{-}} fragments for {CH} units results in the neutral closo-carbaborane.
By determining the number of vertices in a cluster and counting the electron pairs Wade’s rules can be used to predict any cluster geometries. However, the further away from the borane series the cluster type, the more exceptions to the rules are found.
Theoretical Background
2.1 The ab initio/IGLO/NMR method
Nuclear magnetic resonance (NMR) spectroscopy is a powerful tool for the identification of compounds. The ab initio/IGLO/NMR method combines the power of the NMR experiment with a theoretical approach that allows full structural determinations to be made.
Modern NMR machines are built by experts then sold to the end users. The end user is not required to know how exactly the machine works in order to be able to use it. In this sense the device can be seen as a ‘black box.’ The theoretical method is employed in this work using prebuilt commercial and freely available software. This software can also be used with no understanding of the mechanisms involved in the generation of results. But, in order to make full use of the ‘box’, it is important to have an understanding of what is going on inside, to know its strengths and weaknesses.
The three
parts to the method are;
The ab initio step for the computation of theoretical structures.
The IGLO (Individual Gauge for Localised Orbitals) method by which theoretical structures are used to calculate the NMR shift for each atom in all systems.
The experimental NMR evidence is the information that the whole study hinges on. Without this ‘real’ data the method is redundant. The level of agreement between IGLO and experimental values is the criterion by which one competing structural model is preferred over another.
2.2 The ab initio method^{12}
The term ab initio (from the beginning) implies a rigorous mathematical treatment starting from first principles. This is not completely true but the calculations are far more complete and therefore more time-consuming than semi-empirical methods of obtaining chemical data. The method works by deriving approximate solutions to the Schrödinger equation^{13} on the system under study. The Schrödinger equation describes the profile of a simple harmonic three-dimensional standing wave
_{} (1)
In equation (1), y gives the profile of the standing wave associated with a particle of mass m moving in a field of potential V with energy E, where h = h/2p. Ñ^{2}y is the laplacian operator
_{} (2)
Equation (1) is often written in the cryptic shorthand form of
Hy =Ey
(3)
E is the total energy, y is the molecular wave function from which all chemical properties can be calculated and H is the molecular hamiltonian. The complete hamiltonian, H, includes nuclear and electronic kinetic operators, electrostatic interactions between all charged particles, interactions between all magnetic moments due to spin, orbital motions of nuclei and electrons and an accounting for relativistic effects. The resulting hamiltonian is far too complex to work with. The following approximations are used to obtain a simpler version of the hamiltonian.
2.2.1 The Born-Oppenheimer approximation^{14}
Since the nuclei are much heavier than the electrons (the ratio proton mass:electron mass = 1836:1), nuclear motion can be decoupled from electronic motion. This is to say that the frequency of nuclear vibrations is insignificant on the time-scale of electron movement. What this does is to freeze the nuclei in space whilst the electron wavefunctions are mapped around the nuclei.
2.2.2 Relativistic effects
A moving particle experiences a change in mass due to relativity. This effect is experimentally observed for heavy atoms where the core electrons can have velocities approaching the speed of light. To an extremely good approximation, relativistic effects can be ignored for the systems studied during this work as all systems contain no elements heavier than phosphorus.
2.2.3 The Molecular Schrödinger Equation
This is the time independent, non-relativistic Schrödinger equation, the central equation in the ab initio process. Now that the above approximations have been taken into account the molecular hamiltonian takes the general form (in atomic units)
_{} (4)
The first term is the kinetic energy and the second is the Coulomb interaction, a sum of the nuclear-nuclear repulsion, electron-electron repulsion and nuclear-electron attraction.
Within the Born-Oppenheimer approximation, the molecular Schrödinger equation can be factorised into nuclear and electronic wave functions.
y_{total }= y_{electronic} + y_{nuclear}^{ }
(5)
Now that y_{total} has been split into component y_{electronic} and y_{nuclear} terms, the y_{electronic} term can be solved for any nuclear position and then the y_{nuclear} term can be solved on the resulting potential energy surface.
2.2.4 The Electronic Schrödinger Equation
H_{electronic}y_{electronic} = E_{electronic}y_{electronic }
(6)
E is the electronic energy, y is the electronic wave function and H is the electronic hamiltonian (in atomic units)
_{} (7)
The first term only accounts for the kinetic energy of the electrons as the nuclei are static. The second term accounts for electron-nuclei Coulomb attractions and the third term accounts for electron-electron repulsions. The final term takes account of the nuclei-nuclei repulsions, and is constant for any given nuclear configuration.
2.2.5 Electronic Spin
Since electrons are Fermions (particles with a non-integral spin) there are additional constraints on the electronic wave function:
The overall wave function must be anti-symmetric with respect to the exchange of position and spins of any two electrons.
The Pauli exclusion principle dictates that no two Fermions can be in the same quantum state. If y is a function of one-electron functions (orbitals), this leads to the aufbau principle, which explains why electrons occupy orbitals of successively higher energy.
2.2.6 Restricted or Unrestricted?
Once the electron orbitals are factorised into separate spatial and spin components, the principle of anti-symmetry dictates that two spin orbitals with the same quantum number differ only in their spin component.
Spin orbitals in which equality is forced are called restricted orbitals. All systems in this work used the restricted self-consistent field methods in their analysis. A restricted system, having all electrons paired with others of opposite spin, is a closed shell system. In the closed shell wave function the spin components can be factorised out, halving the number of spatial orbitals present.
In unrestricted systems, open shells are present. These include all radical species having unpaired electrons.
2.2.7 Level of theory
The electronic wave function y is now the only unsolvable term left in the electronic Schrödinger equation. Again more approximations are used. The complexity of the approximation of y determines the level of theory.
2.2.8 Basis sets^{15}
A basis set is the set of atomic orbitals built around the static nuclei. The basis set has to be capable of describing the actual wave function well enough to give chemically useful data, but at the same time being solvable in a reasonable time on the computer. The best representation of an atomic orbital is a Slater-type function or Slater-type orbital (STO). These take the general form;
(8)
Gaussian functions in comparison take the general form;
_{} (9)
Fig 2.1 Comparison of Slater and Gaussian type 1s orbitals
A Gaussian function is only 80% true to the structure of an s-type orbital, yet a single exponent gives the energy exactly. Gaussian functions lack cusps at r = 0 and they decay faster at large r than do s-type atomic orbitals. Despite all this, Gaussian type functions are used in the ab initio method.
2.2.9 Why are Gaussians used in preference to Slater-type functions?
The huge advantage of using Gaussian type functions is that their integrals are easily computed. Where the integral of an exponent gives multiple terms, a single Gaussian integrates to another single Gaussian. This has led to the practice of using a number of Gaussian functions to approximate each Slater type orbital. By the use of several values of a in _{ } a set of “primitive” Gaussian functions from compact to diffuse can be derived. The summation of a set of Gaussian functions gives a close approximation to the radial part of the STO as shown for 1s-type orbitals in Fig 2.1.
Fig 2.2 Summation of many 1s type Gaussians approximates a Slater type function.
Other orbitals can also be created by multiplication of the orbital by the standard q and f dependencies (spherical harmonics) to generate p and d shells.
Fig 2.3 Other orbitals can be represented by Gaussian type functions.
Once the set of Gaussian functions is created and optimised their relative dependencies are frozen. Treated as a single function these sets of Gaussian functions are known as a contracted Gaussian function or basis function. Practically, basis sets (a set of basis functions) are obtained from the literature. The basis sets that are used in this work are defined as follows.
STO-3G
Each STO is approximated to a linear combination of three Gaussian primitives.
3-21G
Each inner shell STO is represented by the sum of three Gaussians and each valence shell STO is split into inner and outer parts described by two and one Gaussian respectively.
6-31G
Each inner shell STO is represented by the sum of six Gaussians and each valence shell STO is split into inner and outer parts described by three and one Gaussian primitive respectively.
6-31G* ^{16}
The 6-31G basis set augmented with six d-type Gaussian primitives on each heavy (Z >2) atom, to permit polarisation.
6-31G**
The extra star augments the 6-31G* by adding 3 p functions to every hydrogen atom to permit polarisation.
2.2.10 The Molecular Orbital^{17}
Now that the atomic orbitals are defined by the basis set the process of building the molecular orbital can begin. In molecular orbital theory the wavefunction of the system is the anti-symmetrized product of the one-electron orbitals and every one-electron orbital is a complex linear combination of the atomic orbitals. This shows the three tiers of complexity involved in the process.
The first step in building a molecular orbital is to know the integral of each atomic orbital. This information is then used to build the one-electron integrals. In this step the size of the basis set is crucial. It certainly seems strange to add d functions to second-row elements but their existence cannot be ignored when trying to obtain a close approximation to the true molecular orbital.
Finally, the building of the molecular orbital takes place through the anti-symmetrized summation of the component one-electron orbitals. As all the systems used in this work have paired electrons, two electron integrals are used, halving the computational complexity.
So far, no electrons have been placed into the system but the orbitals they may occupy have been well defined. To know where the electrons go we must calculate the relative energy for every orbital.
2.2.11 The Hartree-Fock Equations^{18}
The Hartree-Fock self-consistent field (SCF) equations are based on the fact that the molecular orbital is a single product of the component one-electron orbitals. The energy of the system is therefore the sum of the one-electron energies (kinetic energies and electron-nuclear attractions) and coulomb interactions between the charge clouds of all pairs of electrons.
The Hartree-Fock equations for a system are derived by finding where this energy is at a minimum. This is done through an iterative process adding different amounts of the atomic orbitals to build the one-electron orbitals, calculating the energy of the resulting set of one-electron orbitals, then summing the energy of the occupied orbitals. Eventually dE = 0 and at this point the SCF has converged.
2.2.12 Correlation Energy^{19}
Correlation energies come about because the HF method considers wavefunctions as products of functions of independent co-ordinates. This corresponds to saying that the probability of finding an electron at position x,y,z is not influenced by the positions of any other electrons. As electrons repel each other this is not true. In reality their motions are correlated. The HF wavefunction results in a higher energy for the system because it has no way of correlating the orbitals. The energy difference between the HF solution and the “exact” solution (for the non-relativistic time independent hamiltonian) for a system is referred to as the correlation energy.
2.2.13 Møller-Plesset Perturbation Theory (MP)^{20}
Rayleigh-Schrödinger many-body perturbation theory as applied to molecular systems by Møller and Plesset and implemented by Pople and co-workers into the Gaussian software, provides a better way of deriving the energy of the system. Is has been implemented from second-order (MP2) through to fifth-order (MP5).
Practically we can only solve MP2 (frozen core), also written as MP2(fc), for the systems studied in this work. Using this level increases computation time at least five-fold. The frozen core means we assume that the inner electrons do not interact with other electrons. This method of SCF derivation can also be used in restricted or unrestricted forms.
Now our level of theory is dependent upon the basis set and which method of SCF determination we use. The notation to define the chosen level of theory used takes the general form:
SCF method/basis set
(e.g. RHF/3-21G means restricted Hartree-Fock using the 3-21G basis set.)
2.2.14 Optimisation
Now we know that for any set of geometries the electronic Schrödinger may be solved, the only thing left to find is the energy minimum. Moving the nuclei of the system will result in a different SCF therefore the system will generally have a different energy. The minima can be found by fitting for example a quadratic expression to geometries of similar energies. Moving the atomic positions closer to the turning point of the quadratic expression will eventually result in the atoms reaching the energy minimum. When such an energy minimum is located in which the nuclei don’t move, or don’t move very much, then this is called the optimised geometry. This geometry may then be passed onto the next stage of the calculation.
2.3 IGLO^{9}
We know that atoms in a molecule that have spin can be observed by NMR spectroscopy. The NMR signals observed can tell us about the local environment of atoms in the molecule. Pascal’s rules show us that magnetic susceptibilities of many diamagnetic molecules can be represented as a sum of transferable atomic and bond properties. Unfortunately, the quantum theory behind magnetic susceptibility leads to very complex expressions that cannot be easily decomposed into localised properties.
Maestro and Moccia discussed the transformation of computed magnetic properties to “approximately” a sum of localised contributions but Kutzelnigg was first to suggest a method to calculate the localised contributions directly.
The practical implementation of the IGLO method was discussed by Meier and van Wüllen in the DIGLO program. The program computes a self-consistent field, then uses the electron map that lies over the nuclear position to predict the NMR tensors. When the program predicts the NMR signals it takes every atom in isolation from the rest of the system and looks at the local orbitals. From this, it gauges the NMR tensor with respect to a standard molecule.
References
1 (a) F. Jones, Chem. News, 1878, 38, 262. (b) F. Jones and R. L. Taylor, J.Chem.Soc., 1881, 39, 213.
2 (a) A. Stock and C. Massanez, Ber., 1912, 45, 3539. (b) A. Stock. Hydrides of Boron and Silicon, Cornell University Press, 1933.
3 R. P. Bell and H. C. Longuet-Higgins, Nature, 1945, 155, 328.
4 W. N. Lipscomb “Boron Hydride Chemistry”, E. L. Muetterties (Ed.), Academic Press, New York, 1975.
5 (a) Partial Retention of Diatomic Differential Overlap (PRDDO/M) is an approximate molecular orbital program that simulates ab initio molecular orbital calculations in a small fraction of the computational time. The original PRDDO method: T. A. Halgren and W. N. Lipscomb, J. Chem. Phys, 1973, 58, 1569. (b) The extension to d orbitals: D. S. Marynick and W. N. Lipscomb, Proc. Nat. Acad. Sci. U.S.A., 1982, 79, 1341.
6 E. A. Laus, R. M. Stevens and W. N. Lipscomb. J. Am. Chem. Soc., 1972, 94, 8699; J. H. Hall Jr., D. S. Marynick and W. N. Lipscomb. J. Am. Chem. Soc., 1974, 96, 770.
7 R. A. Beaudet. “Advances in Boron and the Boranes”, J. F. Liebmann, A. Greenburg, R. E. Williams, (Eds.); VCH Publishers: New York, 1988; Chapter 20, p. 417.
8 (a) M. Bühl and P. von R. Schleyer. “Electron Deficient Boron and Carbon Clusters”, G. A. Olah, K. Wade and R. E. Williams, (Eds.), Wiley Interscience Publication, 1991, Chapter 4, p 113. (b) M. Bühl and P. von R. Schleyer. J. Am. Chem. Soc., 1992, 114, 477.
9 (a) W. Kutzelnigg, Isr. J. Chem., 1980, 19, 193. (b) M. Schindler and W. Kutzelnigg, J. Chem. Phys., 1982 76, 1919. (c) W. Kutzelnigg, U. Fleischer and M. Schindler, “NMR Basis Principles and Progress”, Vol. 23, p. 165. (d) Springer, Berlin Heidelberg, 1990. (e) U. Meier, C. van Wüllen and M. Schindler, J. Comput. Chem. 1992, 13, 551.
10 K. Wade, J. Chem. Soc. Chem. Commun., 1971, 792.
11 G. A. Olah, K. Wade and R. E. Williams. “Electron Deficient Boron and Carbon Clusters”, Wiley Interscience Publication, 1991, Chapter 1, p. 5.
12 (a) W. G. Richards and D. L. Cooper. “Ab initio molecular orbital calculations for chemists”, second edition, Oxford Science Publications, 1983. (b) T. Clark, “A Handbook of Computational Chemistry”, Wiley Interscience Publication, 1985, Chapter 3, p. 93 & Chapter 5, p. 233. (c) J.P. Lowe, “Quantum Chemistry”, second edition, Academic Press, 1993. (d) G.H. Grant and W.G. Richards, “Computational Chemistry”, Oxford Chemistry Primers, Oxford Science Publications.
13 E. Schrodinger, Ann. Physik, 1926, 79, 361.
14 M. Karplus and R. N. Porter, “Atoms and molecules: an introduction for students of physical chemistry”, W. A. Benjamin, New York, 1970.
15 E. R. Davidson, D. Feller, Chem. Rev., 1986, 86, 681.
16 M. J. S. Dewar, B.M. O’Conner, Chem. Phys, Lett., 1987, 138, 141.
17 C. C. Roothan, Rev. Mod. Phys., 1951, 23, 69.
18 V. Fock, Z. Physik, 1930, 61, 126.
19 R. J. Bartlett and J.F. Stanton, “Applications of Post-Hartree-Fock Methods: A Tutorial”, Annual Reviews of Physical Chemistry, Vol 42, H. L. Strauss, G. T. Babback and S. R. Leone (Eds.), VCH publishers. p 615.
20 Knowles, K. Somasundram, N. C. Handy and K. Hirao, Chem. Phys. Lett., 1993, 211, 272.