28 introduction to atomic and molecular structure (i) Find two resonance forms which have no charge separation which are also important in the description of the molecule. (ii) With the help of these resonance structures provide a rationale for the differences in bond lengths measured experimentally for this molecule. 1.6 (i) Give a Lewis structure for cyanamine H2NCN which does not place formal charges on the atoms. What is the spatial geometry for this structure? (ii) Generate another Lewis structure by delocalizing the lone pair on the amino group. What is the geometry associated with this structure? (iii) Experimentally it is found that the amino group is slightly pyramidal and the inversion barrier very small. Explain this result. 1.7 Collect together all the molecules with the formula H4C20 in which the octet rule is satisfied at each atom. (i) Give the Lewis structure and the geometries of the two molecules containing the C—C—O unit. (ii) Do the same for a cyclic molecule. (iii) Give a structure for the molecule containing the C—O—C unit. Show that one cannot be written without charge separation. Complete the description ■ of the molecule knowing that the two CO bonds are equal. 1.8 (i) Show that there exists for the carbon monoxide molecule (CO); (a) A Lewis structure where the octet rule is obeyed at each atom but which contains a separation of charge, and (b) a Lewis structure which docs not have a separation of charge but does not obey the octet rule, (ii) Consider the bond between M and CO in an MCO species where M is a Lewis acid, i.e., contains a vacant site (M for example could be a transition metal). Using the resonance forms found above write down the different bonding schemes for the M—CO bond in the two following cases; (a) M does not contain a lone pair, (b) M has a lone pair able to be dclocalized in addition to a vacant site. 2 Properties of atoms A number of experimental observations have shown that the movement of microscopic particles cannot be correctly described within the framework of classical or Newtonian mechanics. Since the development of quantum mechanics in the 1920s, this tool has become indispensable in understanding phenomena at the microscopic scale, from those associated with the atomic nucleus to those associated with molecules. The cornerstones of this approach require a substantial body of mathematical background but we will limit ourselves to just enough needed for an elementary description of atomic structure. We shall unashamedly sidestep many proofs and concepts which are too complex for this book and refer the interested reader to more specialist books in the bibliography. 2.1. Elements of quantum mechanics Just as in classical mechanics, quantum mechanics possesses its fundamental equation, the Schrodinger equation from which, in principle, most properties may be derived. We will illustrate some of its general characteristics by using the hydrogen atom as an example. This, with its single electron moving around a nucleus comprising a single proton, is surely the simplest system which one can study. 2,1.1. Schrddinger's equation Within the language of quantum mechanics the electron is not described as a point mass associated with a trajectory in space, but rather as a wave represented by the mathematical function *P whose value depends upon the x, y, z coordinates of the space which it occupies. ¥ is usually referred to as the wavefunction. This very non-classical idea makes contact with the notion that atoms consist of a collection of electrons surrounding a nucleus in the following way. The square of this function W2(x, y, z) represents the probability density of finding the electron at this point. (Sometimes VP will turn out to be a complex function containing real and imaginary parts. In this case we need to use "FT* where *P* is the complex conjugate of *P). In other words the probability dP of finding the electron in an infinitesimal volume dr centered around the point given by the coordinates x0, y0, z0 is given by the expression dP = *2(x0, y0, z0) dx (1) 30 introduction to atomic and molecular structure properties Of atoms 31 The probability of rinding the electron somewhere in space has to be unity and so the function "V has to satisfy the relationship (2), where the integration takes place over all space. We say that XV is normalized. x\'\x, y, z) di = 1 (2) J space The function is determined by solution of the Schrodinger equation (3). Here Jf is an operator, called the Humiltonian = £T' (3) which operates on the function "P and transforms it into another function. E is a scalar whose value is equal (in the cases we shall study) to the energy of the electron. Solution of equation (3) consists, therefore, of finding those functions, which after being transformed by the operator Jf, may be written as a product of the original function multiplied by a scalar. Such solutions are called the eigenfunctions of the operator Jf and the scalars with which they are associated, the corresponding eigenvalues. We finally note that from all the possible solutions we only retain those which make physical sense, namely those which are normalized as in equation (2). Solution of the Schrodinger equation amounts to searching for the collection of pairs PF„ Es) which satisfy the relationships (2) and (3). The principles behind the solution of the Schrodinger equation are difficult to illustrate in a simple fashion, but we can give a simple analog from the field of vector geometry. Let us consider an operator (R) which behaves like a planar mirror (2.1) and search for the eigenfunctions (v) and eigenvalues (X) obtained by solution 3-1 of the equation Rt> = Xv. Obviously there are two types of functions. Those (for example vu v2) which lie completely in this plane are transformed by the mirror plane operation into themselves and are therefore associated with an eigenvalue of +1 (i.e., Rl>j =(+1)!>j). Those which lie perpendicular to the plane are transformed into vectors pointing in the opposite direction and therefore have an eigenvalue of — 1. (i.e., R;;3 = ( — l)t>3.) Three solutions of Rti — Xv are therefore (vlt + 1), (v2, +1) and Among the various solutions of the Schrodinger equation, that which corresponds to the lowest energy of the system (4^, fij) has a special importance. This is the state of the system, the ground state, which is the most stable. All the other solutions 0Ff, £,) correspond to excited states. 2.1.2. Some important properties of the eigenfunctions Some of the properties of the eigenfunctions of the Hamiltonian operator will be useful to us in what follows in subsequent chapters. To establish these we need first to state that the operator is linear, that is to say it satisfies the relationships mf(m) = xjccv) (4) + Vj) = + JfOF,) (5) (a) The sign of \ji Let us suppose that is a normalized solution of the Schrodinger equation associated with an eigenvalue £, such that .;'/■' T, • hj'H, (6) with *fdt = i ,' (7) J space Let us consider for the moment the function )y^l, where X is a scalar. Combining the relationships (4) and (6) we get Mf(X%) = X3f(%) = XEt% = Ei(X%) (8) Thus the function XWj is also a solution of the Schrodinger equation and is associated with the same eigenvalue (£;) as *P( itself. This function has to be normalized such that (X%)2dx=i i.e. X1 V,2dT=l (9) J space J space so that X2 — 1 or X— ±1. There are therefore two solutions for X, the first corresponding to the initial function % (X — 1) and the second to its negative, — *F( (a = — i). Thus if v¥l is a normalized solution of the Schrodinger equation, so is its negative, — x¥l, and thus corresponds to the same electronic description as a result. A physical sense may not then be attributed to the sign of the wavefunction. The function yvi may be used as equally well as its negative (b) Overlap and the orthogonality of eigenfunctions The overlap integral of two functions % and ^ is the integral over all space of their product. We use a useful notation, due to Dirac, to write this in a condensed 32 introduction to atomic and molecular structure properties of atoms 33 fashion <F,> = %Vjdr (10) J space where dt is the infinitesimal volume element. (Equation (10) describes the situation for real functions T. For complex functions we need to use the expression J *¥f*¥j dr where P, ¥,> <= 1 (12) Thus the collection of eigenfunctions of the Hamikonian operator form a set of orthonormal functions. An analog might be the set of orthogonal vectors k normalized to be of unit length which are orthogonal and define the x, y and z directions of three-dimensional space. (c) Degenerate solutions Suppose for the present that we have two different solutions % and % which are associated with the same eigenvalue £, i.e., 3f% = £¥, (13) S?"¥j = E% (14) We say that these two solutions are degenerate. Applying the Hamillonian operator to an arbitrary linear combination of them (X% + pif/) and using the relationships of equations (4), (5), (13) and (14), we gel tt(X¥, + fi%~) = XJt(%) + pM'QVj) = XE % + nEVj = E(A% + 11%) (15) Thus all linear combinations of two degenerate eigenfunctions are themselves eigenfunctions of the Hamiltonian, associated with the same eigenvalue, E. 2.2. The hydrogen atom 2.2.1. Solutions of the Schrddinger equation Solution of the Schrbdinger equation for the case of the movement of a single electron under the influence of the nucleus leads to an infinite set of QF„ £,). The two particles, nucleus and electron, interact via a Coulombic, electrostatic type of interaction and, if we were going to solve the equation algebraically, we would insert the kinetic and potential energy into the Hamiltonian to produce (as it turns out) a differential equation which is readily soluble. (a) Allowed values of the energies (£,) and atomic spectra This book is not the place to detail the mathematical solution of the Schrddinger equation, but highlight the results. First, the eigenvalues of the Hamiltonian, Et are all negative and second they are inversely proportional to n1, where n is a positive integer called the principal quantum number. E„=-Ry/n2 n= 1,2,3,... (16) Ry is a constant with units of energy and equal to 13.6 cV*. The allowed values of the energy are therefore E1 = — Ry (n = 1) ground state E2= — Ry/4 (n = 2) first excited state £3 = — Ry/9 (n = 3) second excited slate etc. We will always use the convention that the lowest energy state is the one with the most negative value of the energy. So the electron in the hydrogen atom may not have an arbitrary energy, but one of the possibilities given by equation (16). We say that the energy is quantized. It only depends upon the value of the principal quantum number, n. Suppose that a hydrogen atom finds itself in an excited state (n > 1). The return of the electron to the electronic ground state with n = 1 is accompanied by the liberation of energy AE equal to the energy difference between the ground state and the excited state initially populated. AE = -Ryjn2 - (~Ry/l2) = Ry(l - 1/n2) (17) This change in energy is accomplished via the emission of a photon with an energy equal to A£. From the Planck-Einstein relationship the frequency (v) of the emitted photon is given by A£ = hv (18) * An electron volt (eV) is the energy acquired by an electron on moving through a potential difference of one volt. Although it is not an SI unit, it is frequently used to measure energies on the atomic scale. In numerical calculations its SI equivalent, I eV = 1.602 x 10"19 J should be used (see exercise 2.1). 34 introduction to atomic and molecular structure where h is Planck's constant (h — 6.62 x 10" 34 J s). Thus the frequency of the photon emitted in the present case is given by i.e., hv = Ry{\ - 1/n2) v = Ry/h(l - 1/n2) (19) When the excited state is sufficiently high in energy (n > 2) the electron doesn't necessarily have to return to the ground state (n = 1), but can instead move to another excited state (labelled by ri) of lower energy than the initial one. This implies ri < n. In this case the frequency of the emitted photon is given by the relationship v = Ry/n(l/n'2 - 1/n1) (20) As a general result of such processes one obtains for each value of ri a series of spectral lines which together constitute the emission spectrum of the atom. They may be grouped according to the value of ri and are traditionally named after the physicists who discovered them (Figure 2.f). The Lyman series is a special one in that it corresponds to the return of excited electrons to the ground state with ri = 1. The inverse of emission is absorption. A hydrogen atom in its electronic ground state can absorb a photon, using its energy to move to an excited state. Such an absorption process can only occur if the photon energy corresponds exactly to the energy difference between the excited state and that of the ground state, namely hv = Ry(l - 1/n2) (21) The larger the value of n and therefore the higher in energy the excited state, the higher the frequency of the photon needed for excitation. When n -» oo the energy of the excited stale tends to zero, giving rise to a situation where the electron and nucleus are completely separated from each other. This corresponds to ionization of the hydrogen atom, H -» H+ + e~, and the energy involved, A£m is simply given by A£„ = Ry = 13.6 cV (22) Thus the quantity Ry is the energy needed to ionize the hydrogen atom in its electronic ground state (the ionization potential) and is found experimentally to be just this, 13.6 eV. It is the smallest amount of energy needed to detach an electron from the atom in its ground state. The energy needed to ionize the atom in an excited state can be obtained in the same way. In general the rules which define the allowed frequencies of light Tor photon absorption are just the same as the ones we have described above for photon emission. (b) Nomenclature for the eigenfunctions, T, The principal quantum number n is sufficient to characterize the allowed values of the energy, but the situation is more complex when it comes to a description of properties of atoms 35 E (eV) -13.6 yv v n = 5 n = 4 n = 3 Paschen Balmer i j i Lyman Figure 2.1. Transitions observed in the emission spectrum of atomic hydrogen, the corresponding eigenfunctions. We need three quantum numbers to classify them. (i) The principal quantum number, n, which as we have seen is a positive integer (n = 1, 2, 3,...). b (ii) A secondary quantum number, I often called the angular momentum, or azimuthal quantum number. It is an integer, positive or zero and is always less than n. 0 re[0, oo[ y = rsin flsin

c [0, 2ti[ (a) Analytical form Because of the form of the mathematics of the hydrogen atom problem it turns out that the wavefunctions may be written as a simple product of two functions, one radial in extent and the other containing all of the angular dependence. (25) The first term, the radial one (fl„,,(r)) only dependent on r, contains the principal quantum number n and the angular momentum quantum number /. The second term Yi,m(8> )> the angular part, only depends on the variables 9, . Equations (26) and (27) show this process formally. R2»2 dr = 1 (26) Yl„ (9, ) sin 9 d9 d = 1 (27) As we will see it will prove very useful to use a pictorial representation for the wavefunctions. However, since they in general depend upon the three variables r, 0, and (/>, ii is impossible to rigorously represent their shape in two dimensions. The most satisfactory way is to draw a contour map of the function by drawing lines of constant ¥ in a plane of interest. 38 introduction to atomic and molecular structure properties of atoms 39 (b) The 1 s function = 1, / = 0, m = 0) The analytical expression for this Function is M', 2 .A? exPl - - (28) The expression doesn't appear to contain the variables 0 and 4>. In fact the angular contribution is equal to sJU/4n). The radial part varies via the term exp(-i-/a0) and the expression 2/al'2 assures that this part of the wavefunction is normalized. a0 is a universal constant of length, known as the Bohr radius, equal to 52.9 pm. This wavefunction is said to be spherically symmetrical since its value at a point only depends on the distance of that point to the nucleus. It is thus easy to precisely describe the behavior of the Is function using a plot of its dependence on r, the single variable on which it depends (2-3a). The amplitude of the wavefunction is largest at V,. (O 0,5 0,1 2-3a the nucleus and decreases exponentially via the term in -rja0. It is important to note that the wavefunction has the same sign (positive from equation (26)) for all values of r. The contour diagrams for Vu are easy to construct since the surfaces of constant T are spheres (i.e. constant r) and their intersection with a plane containing the nucleus are circles (2-3b). These two representations arc not very convenient ones in a practical sense. A representation which captures the essence of the wavefunction in terms of spherical symmetry and a sign which doesn't change with r is shown in 2-4a. *Pls is represented 2-4a o 2-4b by a circle centered at the nucleus and containing a positive sign to show that the wavefunction is positive everywhere. An equally valid possibility is one which contains a negative sign, indicative of a wavefunction which is negative everywhere. Recall that from Section 2.1.2a this gives a completely equivalent description of the electron. The convention we will use in this book is shown in 2-4b. Hatching or shading is used to indicate where the wavefunction is positive, and the circle is left empty when it is negative. (c) The 2s function (n = 2, 1=0, m = 0) The analytic expression for this function is I 2--I exp r 2a, 1 (29) Just as for the Is orbital the angular part of the 2s function is also constant and equal to v''(l/47r). Thus the wavefunction depends only on r, and as a consequence is spherically symmetrical. Its amplitude tends to zero as r tends to infinity via the exponential term, just as for the Is function. What is new here however is that the term 2 - r/a0 goes to zero when r = 2a0, and the probability of finding the electron at this distance from the nucleus, described by the surface of a sphere of radius 2a0, is identically zero. Wc say that the 2s function possesses a spherical node. As defined in equation (29) it is positive when is less than and negative when r is greater than 2an. A surface where the wavefunction is zero everywhere on it is called a nodal surface. The wavefunction changes sign on moving from one side to the other. It is again possible to describe the behavior of >P2s as a function of r in a simple way since the wavefunction docs not explicitly contain the variables 9, 4> (2-5a). It 2-5a 2-5b has a maximum at r = 0, changes sign at r = 2a0, as indicated by the change from solid to dashed contour lines (2-5b) and then approaches zero as r becomes large. It is however more convenient to use the representation 2-6 which comprises a pair of concentric circles, containing plus and minus signs, or hatched and unhatched areas to describe the sign of the wavefunction as described earlier. 40 introduction to atomic and molecular structure 2-6 (d) The 2pfunctions (n = 2, /= 1, m= +1, 0, -1) Although the function which describes 2p0 is real, it turns out that the functions describing 2p+1 and 2v-\ are complex conjugates of each other. In general the dependence of Ylm (6, (j>) turns up in the form e'm* leading to values of e1*, 1 and e~'* for p1; pu and p_x respectively. Since 2p + 1 and 2p_: are degenerate, judiciously chosen linear combinations of the two are also valid wavefunctions, as described in Section 2.1.2c. Thus wc could write for one combination (e1* + e~'*)/2 = cos 4> and (c1* — e~'*)/2i = sin for the other. This leads to two new, real, orthonormal functions. Along with the function for 2p0 we now have the three functions exp exp cxp 'Tl 2aJ_ [3 /— sin 6 cos V 4k -)] 2aJ_ F$ ■ a ■ / -sinfl sin V 471 2a0)_ /3 /— cos 0 V 4k (30-a) (30-b) (30-c) These functions take on a simple analytic form by transformation back to the cartesian coordinates of 2-2. The angular part of the three wavefunctions of equation (30) multiplied by r are just the cartesian functions x, y and z. Such a correspondence leads to the following description of the wavefunctions: ^2px = Nx exP f2tt = Ny cxp ¥2l>i = Nz exp r 2a0 r 2a0 r 2a„ (31-a) (31-b) (31-c) where N = - I 2^6(4 V 4tc This new nomenclature emphasizes the similarities which exist between the three properties of atoms 41 functions. Rach possesses the same local geometric properties, they just point along a different goemetrical axis, x, y or z. It is sufficient to take a look at just one of them (p. for example in equation (31-c)) to be able to understand them all. For a given value of z the function 2p. has the same value for all points located at the same distance. ;\ from the origin (2-7a). We say that this function is cylindrical!y symmetrical about the z-axis. On the other hand 2p_ is of opposite sign for two points related by the xy plane, namely zA = — zB and rA = rB (2-7b). This function is thus said to be antisymmetric with respect to the xy plane. Finally 4*2-, is identically zero within the xy plane. This plane is consequently a nodal plane of p.. / z \ Tf A y X / A 2-7a is 2-7b A contour description of the 2p_. function is given in 2-8a. As before the change from solid to dashed contour lines indicates the change in sign of the wavefunction. Generally we prefer the representation shown in 2-Sb, one which contains all of the essential information about the orbital, namely cylindrical symmetry around the z-axis, the presence of a nodal xy plane and a function which is antisymmetric with respect to it. Wc say that this function has two lobes, one positive (shaded in 2-8b) and one negative (unshaded). 2-8a 2-8b 42 introduction to atomic and molecular structure The representations wc use for 2px and 2py can be obtained from 2-8b by changing the axis involved, and are shown in 2-9. Notice the special way we draw the 2px orbital to indicate that it is directed perpendicular to the page. p4^ 2-9 2P, 2P„ 2P, (e) Radial probability density The probability of finding an electron somewhere is an important concept but one which is sometimes difficult to portray. Another way which is frequently used to characterize the wavefunction is the radial probability density which is the probability of finding the electron in the volume enclosed by the two spheres of radii r and r + dr (2-10)*. We need to integrate the square of the function over the angular coordinates, 2-10 0 and r + dr RlMYt«fß, 4>Y sin 6 dr dB d(p dS = = ^,,(r>2dr r,2,„(0, ) sin 6 dO d