MODULE 7

1. MODULE 7 Motion in a Coulomb Field The motion of a particle subject to the restriction of a spherically symmetrical (varies as 1/r) electrostatic (Coulomb) field is centrally important to chemistry. It underpins the structure and spectroscopic properties of hydrogenic atoms (one nucleon + one electron entities such as H, He+ Li++, Be3+, etc). Most of the work necessary for us to analyze this problem has already been done because the motion of a captive particle in a central force field can be thought of as being akin to that of its motion on a nested stack of thin spheres, with the ability to pass between the spheres.

2. MODULE 7 The wavefunction will probably resemble that of the particle on a sphere, with an added term to account for the radial motion.

3. MODULE 7 The Hamiltonian one nucleon + one electron (2 particles in a mutual force field) system where the potential energy term has radial symmetry. subscripts ‘e’ and ‘N’ refer to the electron and the nucleus, respectively. Both particles have motions within their mutual Coulomb field in addition to their translational motion through space. This latter is separate from the inner motion and has no influence on it and is ignored.

4. MODULE 7 Here m is the reduced mass of the system and the motion is considered to be the relative motion of the two particles in the mutual field. In most cases, m ~ me since Unlike the previous case our motion has variable r, so we need the full Laplacian in spherical polar coordinates in our Schrödinger equation.

5. MODULE 7 The potential energy term inside the parentheses on the LHS of the Schr equ depends on r only (it is a centro-symmetric V) This is separable into the radial and angular equations where P= rR, and the quantity in braces is the so-called "effective potential energy" It is made up of the attractive Coulomb term and a repulsive contribution (that depends on l) and can be thought of as being due to the centrifugal force imparted on the electron by virtue of its motion.

6. MODULE 7 Plot of Veffvs.r R is termed the radial wavefunction When l = 0 the repulsive part of Veff is absent and the potential is everywhere attractive. In this case there is a non-zero probability of finding the electron close to the nucleus. When l is non-zero it has the effect of keeping the electron away from the nuclear region. At large r, R decays exponentially towards zero.

7. MODULE 7 Hydrogenic radial wavefunctions Acceptable solutions of the radial equation are the associated Laguerre functions, providing two quantum numbers n and l. where r = (2Z/na0)r and Allowed values of n are 1, 2, 3, … and of l are 0, 1, 2, …, n-1

8. MODULE 7 Note that the functions with l = 0 (1s, 2s, 3s) are non-zero and finite at r = 0, those functions with l > 0 are zero at r = 0. Some functions have nodes (zero crossings). Each radial function has (n-l)-1 nodes. The locations of the nodes are determined by equating the polynomial part of the wavefunction to zero. Note (see Table) that Rnl is always real

9. MODULE 7 For example for the y3,0 function we set Putting the radial wavefunctions into

10. MODULE 7 E is independent of l or ml. Thus in hydrogenic atoms (only) the state energy depends only on the principal quantum number, n, and each level is n2-fold degenerate. Figure depicts the values of Envs. n. They are shown as multiples of hcR, where R (Rydberg constant) is the value of the parenthetic term in equation above expressed in cm-1.

11. MODULE 7 The bound states have negative energy values. Zero is the energy of an electron-proton pair at r = infinity. The shaded region represents the classical energy continuum for an electron-proton pair. It is in contrast to the quantum-derived discrete set of energy states, depending on n2.

12. MODULE 7 The continuum region at positive energies represents other solutions of the Schrödinger equation. These are the unbound states to which an electron is raised when it is ejected from the atom by high-energy interactions Since unbound the electron is a free particle, its motion is not quantized, and its energy is a continuum. The energy differences between the bound states can be computed using equation. Energy values that closely match spectroscopic values.

13. MODULE 7 Total wavefunctions We saw that the Schrödinger equation was separable into r-dependent and q, f -dependant parts Thus the total wavefunction for a hydrogenic atom is the product of the appropriate spherical harmonic (Ylml) with the appropriate radial function Rn,l. So to build the function for given values of n, l, and ml, simply write down the relevant Ylml from Table 6.1, and multiply it by the appropriate Rn,l from Table 7.1.

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15. MODULE 7 Probabilities and radial distribution functions The probability of finding an electron in a hydrogenic atom within a volume element dt at a point specified by the polar coordinates, r, q, f is given by This enables us to evaluate the "point probability", i.e. that at a specific, sharply defined point along a given r direction. However it is often useful to know the probability of finding the electron at a given distance from the center, regardless of direction. This can be obtained by integrating over the volume contained between a pair of concentric spheres of radius r and r+dr

16. MODULE 7 [in spherical polar coordinates dt = r2sinqdqdfdr] The spherical harmonics are normalized to 1 in the sense that P(r) is called the radial distribution function. When multiplied by dr, it provides the probability that the electron will be found between r and r+dr. [any direction]

17. MODULE 7 For the state I1,0,0> using Table 7.1 Note that the wavefunction itself (R) is an exponential declining from a non-zero value at r = 0, so the point probability at the nucleus (R2) is also non-zero The r2 factor forces P(r) to zero at r = 0 P(r) tends to zero at large values of r because the exponential term has a larger influence than the r2 term

18. MODULE 7 To find P(r)max we differentiate the function wrt r and set the derivative equal to zero. For the function shown (a0 = 52.9 pm). Thus, we see that as Z increases, rmax decreases because the electron is drawn closer to the nucleon as its charge increases. Note that for orbitals with l = 0, R2r2 is equivalent to 4pr2y2.

19. MODULE 7 ATOMIC UNITS A new system of units adds convenience and brevity. All eigenvalue equations, eigenfunctions and eigenvalues are simpler to write down. For hydrogenic atom

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21. MODULE 7 Atomic Orbitals A one-electron wavefunction for an electron in an atom is termed an atomic orbital. All hydrogenic spatial atomic orbitals (one-electron wavefunctions) are defined by the three quantum numbers, n, l, and ml. In such a situation we say that the electron "occupies" the orbital that is described by the particular one electron function. We can use the ket notation to describe the orbital For example the ket I1,0,0 > denotes an orbital that has n = 1, l = 0, and ml = 0.

22. MODULE 7 For historical reasons that have origins in experimental spectrometry Atomic orbitals (AOs) with l = 0 are termed s-orbitals, with l = 1, are termed p-orbitals, with l = 2, are termed d-orbitals, with l = 3, are termed f-orbitals. An electron that occupies an s-orbital is called an s-electron, and so on.

23. MODULE 7 The orbitals for a given value of n are said to define a shell of the atom. A commonly used terminology calls the n = 1 shell as K, the n = 2 shell as L, the n = 3 shell as M, and so on. Atomic orbitals with the same value of n, but differing in their values of l are said to form sub-shells of a given shell. In hydrogenic atoms, all AOs of a given shell (same principal quantum number) are degenerate Read more about shells in Module 7

24. MODULE 7 Atomic orbitals can be represented by boundary surfaces, within which 90% of the probability density is contained As depicted in Figure (left) the s-orbital, with l = 0, has spherical symmetry because when l = 0 the spherical harmonic contains no term in q or f. However for the p-orbital l = 1, and the spherical harmonic, depending on the value of ml contains terms in q and/or f

25. MODULE 7 The quantized AM of the particle distorts the symmetry of the AO from spherical The radial functions have rl dependence Thus in p-, d-, and f-orbitals, the electron has an increasing tendency to stay away from the nucleus (Figure)

26. MODULE 7 The three p-orbitals are distinguished from each other by the value that mlcan take for l = 1 (ml= +1, 0, -1) the different ml values provide the components of angular momentum around an arbitrary z-axis. For ml = 0, the z-component has zero amplitude. Y1,0 has angular variation that depends on cosq Then cos2q (the probability density) has its maximum values at q = 0o and 180o, on either side of the nucleus.

27. MODULE 7 For the 2p orbital with ml = 0 the total wavefunction is where f(r) is a function of r only. In spherical polar coordinates, r cosq = z and the AO, in common with all p-orbitals with ml = 0, is termed the pzorbital.

28. MODULE 7 The pzorbital has zero amplitude in the xy-plane (a nodal plane) and the wavefunction changes sign on going from one side of the plane to the other

29. MODULE 7 Two other 2p orbitals are defined with ml=+1 and -1 Functions with this f dependence have AM about the z-axis The functions are complex and to obtain something to sketch it is usual to take the real linear combinations

30. MODULE 7 Note that in spherical polar coordinates rsinqcosf = x, and rsinqsinf = y The px and py orbitals have the same shape as the pz orbital but they are directed along the x- and y-axes, respectively and have nodal planes See Module 7 for d-orbital discussion

31. MODULE 7 Spin Angular Momentum The Schrödinger equation approach is successful at predicting/explaining the positions of spectral lines, particularly for the hydrogenic ions. One phenomenon that eluded the Schrödinger prediction was a closely spaced pair (a doublet) of lines at 589.59 nm and 588.99 nm in the atomic spectrum of sodium atoms. Theory predicted a single line at 590 nm.

32. MODULE 7 Dutch physicists Uhlenberg and Goudsmit (1925) suggested that an electron behaves like a spinning top having z-components of an intrinsic spin angular momentum of magnitude This added a fourth quantum number s to the n,ℓ, and ml quantum numbers that specify the energy of an atomic orbital. s determines the magnitude of the spin AM in the same way that ℓ determines the magnitude of the orbital AM. ms is the z-component of the electron spin angular momentum

33. MODULE 7 The unique value of s is 1/2 and so ms = s, s-1, …, -s = 1/2 and -1/2 Thus the z-component can point to positive values (clockwise rotation) or to negative values (anticlockwise rotation) These alignments are called a (spin-up) and b (spin-down) As with orbital AM the spin vector does not point to a particular direction in space, but it lies somewhere on the cone, the half angle of which is determined by ms, and the vector magnitude

34. MODULE 7 Current thinking dismisses the "spinning top" model of the electron The concept of the electron having an intrinsic angular momentum is retained owing to relativistic extensions to quantum theory developed by Dirac in the 1930s The orbital AM operators have been defined earlier Analogously for spin AM we have

35. MODULE 7 The eigenfunctions a and b are defined by the equations: Just as we could evaluate the magnitude of the orbital angular momentum as Analogously we have and the unique value of S (s = ½) is

36. MODULE 7 Recall that ℓ can become very large, and in the limit its z-component lies along the z-axis (~ classical behavior). This is not possible with spin AM (limited by s = 1/2) thus the intrinsic spin AM of an electron has no classical counterpart-it is strictly a quantum concept a and b are known as spin eigenfunctions. The two spin operators are Hermitian (real EVs) and a and b are orthonormal functions

37. MODULE 7 s is the spin variable (no classical analog) The electron now needs four quantum numbers to specify it and we need to add a spin function to the spatial function (yn,l,ml) We postulate that the spin and spatial parts of the total wavefunction are independent and we can write:

38. MODULE 7 The complete (spatial and spin parts) one electron wavefunction yis termed a spinorbital. For example the two spin orbitals for a 1s electron in a hydrogenic atom are designated as The 100 spatial parts are normalized but the two spinorbitals are orthogonal because the spin parts are orthogonal

39. MODULE 7 Thus four quantum numbers are required to completely specify an electronic wavefunction The Pauli principle requires that no two electrons in an atom can have the same set of four quantum numbers. This is a special case of the Exclusion Principle that states: "All electronic wavefunctions must be ANTISYMMETRIC under interchange of any two electrons." Or i.e. the wavefunction changes sign when electrons i and j are interchanged (more on this later)

40. MODULE 7 Spin-orbit Coupling Electrons in atoms (hydrogenic e.g.) possess AM in two types 1. Arising from the orbital motion 2. Arising from their intrinsic spin These momenta couple through a vector sum The ℓ and s vectors couple according to the sign of ms. The total angular momentum j = ℓ+s, or j = ℓ-s

41. MODULE 7 For hydrogenic ions (single electron) these are the only two possible values of j. For s-orbitals (l = 0), the only permitted value of j = ½ (never negative). As ℓ and s, so j has an z-component quantum number, mj The j-states differ in energy slightly It is this spin-orbit coupling that is responsible for the doublet in the emission spectrum of sodium atoms, referred to above, and for many other effects that we shall examine this topic in more detail later