Applications to Molecular Science

Abstract

CW-ESR spectroscopy combined with matrix isolation method and ionizing radiation is applied to investigate structure and reactions of intermediate radicals in low temperature solid matrices. ESR parameters are predicted with considerable precision by computations, affording a valuable bridge between experiment and theory. Cyclic-C_nF_2n ^– radicals (n = 3–5) have a planar structure with an entirely delocalized singly occupied MO. Unsaturated C_nF_2n-2 ^– radicals have a distorted pyramidal structure occurring by mixing the π^* and σ^* orbitals. The acetylene anion has a trans-bent structure. Structural distortion occurs in methane cation from original T _d to C _2v symmetry due to the Jahn-Teller effect. A similar symmetry lowering from D _3 h to C _2v is discussed for the trimethylenemethane cation. High-resolution ESR spectra of D-labelled methyl radicals in solid Ar are discussed in terms of nuclear spin-rotation couplings. The formation of “H—CH₃” radical pairs and of a “H—H₂” complex” in solid Ar is presented. The structure of [H₂(H₂)H₂]⁺ formed in para-H₂ is discussed based on the ESR spectra of D-substituted samples and on computations.

Appendices

Appendices

1.1 A5.1 A Brief Summary on Quantum Chemical Computation Methods

1.1.1 A5.1.1 Schrödinger Equation

The non-relativistic time-independent Schrödinger equation (SE) [104] for N nuclei and n electrons is:

$$H\Psi = E\Psi$$

((5.16))

where E is the energy of state described by the wave function (WF), Ψ, which depends both on the electronic and nuclear coordinates. Hamiltonian H can be written in atomic units as:

$$\begin{aligned}&H = - \sum\limits_{i = 1}^n{\frac{1}{2}}\nabla _i^2 - \sum\limits_{A = 1}^N{\frac{1}{{2M_A }}}\nabla _A^2 - \sum\limits_{i = 1}^n{\sum\limits_{A = 1}^N{\frac{{Z_A }}{{r_{iA}}}}}\\ &\qquad + \sum\limits_{i = 1}^n{\sum\limits_{j \ge i}^n{\frac{1}{{r_{ij}}}}}+ \sum\limits_{A = 1}^N{\sum\limits_{B \ge A}^N{\frac{{Z_A Z_B }}{{r_{AB}}}}} \end{aligned}$$

((5.17))

The first two terms in Eq. (5.17) are kinetic energy (KE) operators of the electrons and of the nuclei while the final three terms are the operators of electron-nuclear, electron-electron, and nuclear-nuclear potential energies (PEs). M _A is the mass of nucleus A divided by the mass of the electron and Z _A is the atomic number of nucleus A. Terms r _ij, r _iA, and r _AB are the distances between electrons i and j, between electron i and nucleus A, and between nuclei A and B, respectively.

The quantum chemical methods applied to the molecular science such as structural chemistry and chemical reactions described in this chapter are all based on the Born-Oppenheimer approximation [105]. Within the BO approximation the KE operators for nuclei are removed from the above Hamiltonian and the SE is solved for a fixed set of nuclei. The SE within the BO approximation is usually referred to as the electronic SE, and the resulting WFs as electronic states.

The electronic SE is impossible to solve analytically for systems containing more than one electron and further approximations are required. There are two methods available in modern quantum chemistry, i.e. wave function theory (WFT) and density functional theory (DFT) [18–21]. The WFT has been developed to compute an optimal WF for the system, whereas the DFT has been developed to find the optimal electron density (ED) for the system and has becomes an overwhelmingly popular method.

1.1.2 A5.1.2  Molecular Orbitals

A molecular orbital (MO ) has a form of:

$$\Phi_i \left( r \right) = \sum\limits_{\mu = 1}^K{c_{i\mu }\chi _\mu }\left( r \right)$$

((5.18))

where $\chi _\mu$ is an atomic orbital (AO) of atom $\mu$ and the sum extends over all the valence orbitals of all atoms in a molecule. The coefficients of $c_{i\mu }$ can be evaluated by setting up the secular equations and the secular determinant. That is, the associated energies can be evaluated by solving the secular determinant, and then the coefficients are found by inserting these energies into the secular equations. The bond lengths and bond angles of the molecule can be predicted by calculating the total energy for a variety of nuclear positions and then identifying the conformation that corresponds to the lowest energy.

The π MO energies of conjugated molecules can be evaluated using a set of approximations suggested by E. Hückel [106]. In the Hückel method (HM) the π orbitals are treated separately from the σ orbitals, and the latter forms a rigid framework that determines the geometrical structure of the molecule. All the carbon (C) atoms are treated identically, so all the Coulomb integrals for the AOs that contribute to the π orbitals are set equal. The first level of sophistication is the extended Hückel method (EHM), which is a significant extension of the basic HM to include σ orbitals [107]. Current versions of EHM are easy to implement with a computer and a give useful qualitative pictures of MOs in molecules with known structure. Many of the difficulties associated with EHM have been overcome by more sophisticated theories that not only calculate the shapes and energies of MOs, but also predict with reasonable accuracy the structure and reactivity of molecules.

1.1.3 A5.1.3  Hartree-Fock Equations

The starting point for the WFT is provided by Hartree-Fock (HF) theory . In the HF theory the n-electron WF of a molecule is written as a determinant referred to as a single Slater determinant (SD) , ΨSD, constructed from n occupied one-electron WFs, $\Phi_{i} (x):$

$$\Psi_{SD}= \frac{1}{{\sqrt{n!}}}\left|{\begin{array}{ccccccccc}{\Phi _1 \left({x_1 }\right)}&{\Phi_2 \left({x_1 }\right)}&{\begin{array}{ccc}\cdot & \cdot & \cdot \\ \end{array}}&{\Phi _n \left({x_1 }\right)}\\ {\Phi_1 \left({x_2 }\right)}&{\Phi_2 \left({x_2 }\right)}&{\begin{array}{ccc}\cdot & \cdot & \cdot \\ \end{array}}&{\Phi_n \left({x_2 }\right)}\\ \begin{array}{l}\cdot \\ \cdot \\ \cdot \\ \end{array}& \begin{array}{l}\cdot \\ \cdot \\ \cdot \\ \end{array}&{\begin{array}{ccc}\cdot &{}&{}\\ {}& \cdot &{}\\ {}&{}& \cdot \\ \end{array}}& \begin{array}{l}\cdot \\ \cdot \\ \cdot \\ \end{array}\\ {\Phi_1 \left({x_n }\right)}&{\Phi_2 \left({x_n }\right)}&{\begin{array}{ccc}\cdot & \cdot & \cdot \\ \end{array}}&{\Phi_n \left({x_n }\right)}\\ \end{array}}\right|$$

((5.19))

The initial term of $\frac{1}{{\sqrt{n!}}}$ is a normalization factor. The $\Phi _i (x)$ function is a product of a spatial molecular orbital, $\Phi_i (r)$, and a spin function, $\alpha\left(\omega\right)$ or $\beta\left(\omega\right)$. According to the variation principle the best WF is the one that minimizes the total energy$\left\langle{\Psi\left|H\right|\Psi}\right\rangle$ and the optimal MOs can be found by solving the set of n one-electron canonical HF equations:

$$f(i)\Phi (x_i ) = \varepsilon \Phi (x_i ),\quad i = 1\ ...\ n,$$

((5.20))

where ɛ is the energy of the MO, Φ occupied by electron i, and f(i) is the so-called Fock operator:

$$f(i) = h(i) + v{^{HF}}(i)$$

((5.21))

where, the first term, h(i), is the one-electron operator:

$$h(i) = - \frac{1}{2}\nabla_{i}^2 - \sum\limits_{I = 1}^N{\frac{{Z_I }}{{r_{iI}}}}$$

((5.22))

which consists of the kinetic energy operator for electron i, and the potential energy between this electron and fixed nucleus I with atomic number $Z_I$. The second term, $v^{HF}(i)$, is the so called HF potential operator and can be written as:

$$v{^{HF}}(i) = \sum\limits_j{\{J{_j} (i) - K{_j} (i)}\}$$

((5.23))

Two terms in the sum, J _j(i) and K _j(i), correspond to the Coulomb interaction between electron i and all other electrons j, and the exchange interaction between electrons with the same spins, respectively. The calculations of the HF potential energy for electron i require prior knowledge of all other occupied MOs. We have to guess the initial form of the MOs, use them in the definition of the Coulomb and exchange operators, and solve the HF equations iteratively in a process known as the self-consistent field (SCF) procedure.

The spin parts of the MOs in the HF equation, Eq. (5.20), are integrated out. This transform the HF equations to a set of equation involving only the spatial parts. When one forces a pair of α and β electrons to occupy the same spatial part, i.e. closed-shell species, the procedure is called the restricted HF (RHF) . On the other hand, if the α and β electrons are associated with different spatial parts, the procedure leads to the unrestricted HF (UHF) equations, which are employed for radical species. The UHF has an advantage over the RHF to allow for a proper account of spin polarization, a feature which is very important for the study of, for example, hyperfine interactions of radicals. However, a disadvantage of UHF is that the corresponding WF is not an eigenfunction of the total spin operator for the electrons. This means that UHF WFs may be contaminated with spin states of higher multiplicity. One way of measuring the quality of an UHF calculation is to compare the computed expectation value of S ² with the theoretical one.

1.1.4 A5.1.4  Semi-Empirical and Ab Initio Methods

There are two methods, semi-empirical and ab initio methods, for continuing the quantum chemical calculations. In the semi-empirical method , many of the integrals are estimated by using spectroscopic data or physical properties. On the other hand, in the ab initio method , all the integrals in the secular determinant are attempted to be directly calculated.

The Fock matrix has elements that consist of integrals of the following form:

$$\left\langle\chi_{i}\chi_{j}\left|\chi_{k} \chi_{l}\right.\right\rangle = \int\chi_{i}^{\ast}(1)\chi_{j} (1)\frac{1}{r_{12}}\chi_{k}^{\ast}(2)\chi_l(2)d\tau_1 d\tau_2$$

((5.24))

where $\chi_{i}^{\ast}(1)$, $\chi_{j} (1)$ and $\chi_k^{\ast}(2)$, $\chi_l (2)$ are AOs, which in general may be centered on different nuclei. One severe approximation is called CNDO (Complete Neglect of Differential Overlap), in which all integrals are set to zero unless $\chi_{i}^{\ast}(1)$ and $\chi_{j}(1)$ are the same orbitals centered on the same nucleus, and likewise for$\chi _k^\ast (2)$and$\chi _l (2)$. The more recent semi-empirical methods make less severe restrictions about which integrals are to be ignored, but they are all descendants of the CNDO method. These procedures are now readily available in commercial soft-ware packages such as MOPAC [16] and AMPAC [17] which enclose INDO (Intermediate Neglect Differential Overlap), MINDO (Modified Intermediate Neglect Differential Overlap), MNDO (Modified Neglect of Differential Overlap), MNDO/d, AM1 (Austin Model 1), PM3 (Parametric Method 3), SAM1 (Semi-Ab initio Method 1), etc. Here the objective is to use parameters to fit experimental heats of formation, dipole moments, ionization potentials, and geometries.

Commercial packages are also available for ab-initio calculations [21b]. The task to evaluate the integrals is generally facilitated by expressing the AOs used in the LCAOs as linear combinations of Gaussian orbitals. A Gaussian type orbital (GTO) is a function of the form exp(-αr ²). The advantage of GTOs over the fundamentally more correct Slater orbitals (which are proportional to exp(-αr)) is that the product of two Gaussian functions is itself a Gaussian function that lies between the centers of the two contributing functions. In this way the four-centre integrals of $\left\langle\chi_{i}\chi_{j}|\chi_k\chi_l\right\rangle$ become two-centre integrals. Integrals of this form are much easier and faster to evaluate numerically than the four-centre integrals obtained with Slater orbitals.

1.1.5 A5.1.5 Density Functional Theory

The density functional theory (DFT) methods [18–21] focus on finding the optimal electron density (ED) . The “functional” comes from the fact that the energy, and all other properties, of the system are computed as a function of the ED and the ED is itself a function of position (r), ρ(r), and in mathematics a function of a function is called a “functional”. Its advantages include less demanding computational effort, less computer time, and, in some cases (particularly d-metal complexes), better agreement with experimental values than is obtained from HF procedures.

Kohn and Sham (KS) [20] developed a scheme for optimizing the density given an approximate form of this functional, which is known as the KS scheme and underlies virtually all DFT methods used today. The KS scheme is orbital based and starts by introducing a non-interacting n electron system moving in an external potential, $v_s$. Such a system is described by a single SD and optimal orbitals are given by the following equations:

$$\left\{-\frac{1}{2}\nabla^2 + v_s(r)\right\}\Psi_{j}(r) = \varepsilon_{j}\Psi_{j}(r)$$

((5.25))

The ED is constructed from the orbitals by:

$$\rho (r) = \sum\limits_j^n{\left|{\Psi _j (r)}\right|}^2$$

((5.26))

The KS theory proposed a separation of the exact unknown energy functional, $E[\rho ]$, into four parts in terms of orbitals:

$$\begin{aligned}&E[\rho(r)]=-\frac{1}{2}\sum_{j}^{n}\left\langle\psi_{j}(r)\left|\nabla^{2}\right|\psi_{j}(r)\right\rangle-\sum_{j}^{n}\int\sum_{A}^{N}\frac{Z_{A}}{|r-R_{N}|}\left|\psi_{j}(r)\right|^{2}dr\\ &\qquad\quad\quad+\frac{1}{2}\sum_{j}^{n}\sum_{j^{\prime}}^{n}\iint\left|\psi_{j}(r)\right|^{2}\frac{1}{|r-{r^{\prime}|}}\left| \psi_{j}^{\prime}(r^{\prime})\right|^{2}drd{r^{\prime}}+E_{xc}[\rho(r)] \end{aligned}$$

((5.27))

where the first term is the functional for the kinetic energy of the system of non-interacting electrons, the second and third terms are functionals for electron-nucleus and electron-electron Coulomb interactions, respectively. The final term, $E_{XC}[\rho ]$, is the exchange-correlation (XC) functional and is defined as to contain all of what is unknown including the non-classical effects of both exchange and correlation. By applying the variation principle to Eq. (5.27) the set of orbitals that minimizes the energy has to fulfill the following equations:

$$\left[{- \frac{1}{2}\nabla^2 - \sum\limits_A^N{\frac{{Z_A }}{{\left|{r - R_A }\right|}}}+ \int{\frac{{\rho (r^{\prime})}}{{\left|{r - r^{\prime}}\right|}}}dr^{\prime}+ \nu _{XC}(r)}\right]\psi_{j} (r) = \varepsilon_{j} \psi_{j}(r)$$

((5.28))

where $\nu _{XC}(r)$ is the so called XC potential defined by the functional derivative of the $E_{XC}\left[{\rho (r)}\right]$:

$$v_{XC}(r) ={{\partial E_{XC}\left[{\rho (r)}\right]}\mathord{\left/{\vphantom{{\partial E_{XC}\left[{\rho (r)}\right]}{\partial \rho (r)}}}\right. \kern-\nulldelimiterspace}{\partial \rho (r)}}$$

((5.29))

The last three terms in the Hamiltonian in Eq. (5.28) define an effective one-body potential, v _eff(r), which transforms the density of the non-interacting system into the real density. Then, by choosing $v_s (r) = v_{eff}(r)$ in Eq. (5.25) the effective potential is found. As was the case with HF, the one-electron Hamiltonian in the KS equations is solved iteratively and self-consistently. If the exact expression for $E_{XC}\left[{\rho (r)}\right]$ was known, the KS equations would provide the exact non-relativistic ground state solution within the space spanned by a given basis set, including all electron correlation effects. The latter are missing in HF. This is an important difference between HF and KS.

For details of the WFT and DFT the reader is referred to the books by Szabo and Ostlund [18], and Parr and Yang [19]. Furthermore, the book by Koch and Holthausen [20] is recommended as an introductory reading for chemists starting to carry out quantum chemical calculations. Ph.D theses by T. Fängström [108] and D. Norberg [109], and Atkins’s physical chemistry book [86b] served as the main sources to the present appendix on Quantum Chemical Methods: Sections A5.1.1 and A5.1.5 [109], Section A5.1.3 [108, 109], and Sections A5.1.2 and A5.1.4 [86b].

1.2 A5.2 A Brief Summary on Molecular Symmetry

This appendix briefly summarizes “molecular symmetry” (symmetry elements, point groups and character tables) of a molecule so as to facilitate readers’ understanding of the molecular structures and molecular orbitals presented in this chapter. We use some specific examples such as NH₃ with C _3v symmetry, CH₃ radical with D ₃ symmetry (see Fig. 5.18) and c-C₄F₈ ^– radical with D _4 h symmetry (Fig. 5.3) to introduce and illustrate some important aspects of the molecular symmetry.

1.2.1 A5.2.1  Symmetry Elements and Symmetry Operations of Molecules

The methyl radical, CH₃, with a planar trigonal structure, looks the same if it is rotated by 120, 240, or 360° about an axis (C ₃ axis in Fig. 5.18) perpendicular to the plane containing three equivalent carbon atoms. An operation that leaves a molecule (object) looking the same (or sending into itself or a position indistinguishable from the original) is a symmetry operation. Symmetry operations include seven symmetry elements in Table 5.6 which are commonly possessed by molecular systems.

Table 5.6 Symmetry elements and symmetry operations

1.2.2 A5.2.2  Point Groups

A point group consists of all possible symmetry elements possessed by a given molecule. There is always one point, at which an atom is not necessarily present, in the molecule that remains unchanged by the operation. In contrast to this, when we consider crystals, we meet a space group containing operations which displace the molecule to another position in space. Some points groups of chemical interest are listed in Table 5.7 with examples of molecules.

Table 5.7 Some point groups with examples of molecules

A molecule without any symmetry except the identity operation E, such as CHFClBr, belongs to the group C ₁. The NH₃ molecule has one E element, two C ₃ elements (2C ₃) and three σ _v elements (3σ _v) which reflect the molecule through three vertical planes containing the C ₃ axis and it belong to the group C _3v. A molecule possessing an n-fold principal (C _n) axis and n twofold axes perpendicular to the C _n axis belongs to the group D _n. For example, the CH₃ radical belongs to the group D ₃ since it has the E, C ₃, and 3C ₂ symmetry elements. If a molecule possesses a horizontal mirror plane (σ _h) in addition to the symmetry elements of the group D _n it belongs to the group D _nh. For example, c-C₄F₈ ^– radical has the E, C ₄, 4C ₂, and σ _h symmetry elements and it belongs to the D _4 h group.

1.2.3 A5.2.3  Character Tables

A character table is a two-dimensional table whose columns correspond to symmetry operations of the group. Character tables for the point groups C _2v, C _3v, D ₃ and D _4 h are reproduced in Table 5.8. For example, the columns in the C _3v character table are headed by the E, C ₃, and C ₂ operations (Table 5.8b). The numbers multiplying each operation are those of members of each class. That is, two threefold rotations (2C ₃; clockwise and counter-clockwise rotations by 360°/3) belong to the same class. The three reflections (3σ _v; one through each of the three vertical mirror planes) also belong to the same class. The two reflections (σ _v and σ _v ^′) of the group C _2v, however, fall into different classes, see Table 5.8a. Although one can not be transferred into the other by any symmetry operation of the group they belong to the same group by a rule of group theory: the product of any two columns of a character table must be a column in that table. The number of symmetry operations in a group is called its order (“h” in Table 5.7).

Table 5.8 Character tables for C _2v, D ₃ and D _4 h point groups are given as representative one. More character tables are to be found in the textbooks cited as references [86c, 110, 111]

Rows in character tables correspond to symmetry properties of the orbitals, more formally, irreducible representations of the group. The irreducible representations are labelled with large Roman letters such as A₁ and E, but the orbitals to which they apply are labelled with small italic equivalents: for example, an orbital of A ₁ symmtry is called an a ₁ orbital.

The entries consist of characters, the trace of the matrices representing group elements of the column’s class in the given row’s group representation. The character tables of point groups can be used to classify molecular orbitals that belong to the various atoms in a molecule by referring to the different symmetry types possible in the point group. For example, the characters in the rows labelled A and B and in the columns headed by symmetry operations other than the identity E indicate the behavior of an orbital under the corresponding operations: a “+1” indicates an orbital unchanged, and a “–1” indicates that it changes sign. Thus, one can identify the symmetry label of the orbital by comparing the result of changes that occurs to an orbital under each operation with the entries, “+1” or “–1”, in a row of the character table for the point group concerned. The character of the identity operation E tells us the degeneracy of the orbitals. For example, the character in the row labelled E or T refers to the sets of doubly or triply degenerated orbitals, respectively.

In column III in character tables we see six symbols: x, y, z, R _x, R _y, R _z. The first three represent the coordinates x, y and z, while the R’s stand for rotations about the axes specified in the subscripts. In column IV the squares and binary products of coordinates are classified according to their transformation properties. For example, the pair functions xz and yz in C _2v must have the same transformation properties as the pair x, y, since z goes into itself under all symmetry operations in the group.

The textbooks by Atkins and Paula [86c], McQuarrie and J.D. Simon [110], and Cotton [111] served as the main sources for the present appendix on molecular symmetry.