ONE AND MANY ELECTRON ATOMS Chapter 15
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1 See Week 8 lecture notes. This is exactly the same as the Hamiltonian for nonrigid rotation. In Week 8 lecture notes it was shown that this is the operator for Lˆ 2, the square of the angular momentum. ONE AND MANY ELECTRON ATOMS Chapter 5 Hamiltonian for the Hydrogen Atom (5.) Coulomb s law gives the force of attraction between two opposite charges, here the nucleus of charge +Ze and the electron of charge e as F Ze2 r 2 Ze 2 4πε r 2 (gaussian units) (SI units) where ε J C 2 m is the permittivity of free space and e Cisthe elementary charge. For simplicity we will use gaussian units and when numbers are actually required replace every e by e/ 4πε. Since F dv /dr we obtain the spherically symmetric Coulomb potential upon integration, V Ze 2 /r. But for the Hamiltonian in Cartesian coordinates we must replace r by x 2 + y 2 + z 2. Then, separating out the center of mass motion, the Hamiltonian for the internal motion of the hydrogen atom is where, as usual, Ĥ ˆT + ˆV Ĥ h2 2µ 2 x y z 2 Ze 2 [x 2 + y 2 + z 2 ] /2 The square root of the Cartesian coordinates insures that the above Hamiltonian is not separable. Therefore switch to spherical polar coordinates (x, y, z) (r, θ, φ). Since the proton is 836 times more massive than the electron we can replace the reduced mass µ by the mass of the electron m as our text does. The Schrödinger equation becomes 2m h2 r 2 r r2 r + r 2 sin θ θ sin θ θ + r 2 sin 2 θ 2 φ 2 Ze2 ψ (r, θ, φ) Eψ (r, θ, φ) (5.2) r If we return to rotation in three dimensions which we examined in Week 8, we notice that everything in the square brackets except the potential term was in the Hamiltonian for nonrigid rotation in 3D. We only examined rigid rotation where r was fixed. However, the angular dependence (the θ and φ motion) is exactly the same. This suggests that the hydrogen atom Hamiltonian is separable. To see that this is so multiply through by 2mr 2 and rearrange h 2 r r2 r h2 sin θ θ sin θ θ + sin 2 θ 2 φ 2 Ze 2 2mr2 + E r ψ (r, θ, φ) The terms in r have been separated from the angular variables. Furthermore recognizing that ˆL 2 h 2 sin θ θ sin θ θ + sin 2 θ 2 φ 2 we can assume a product form for the eigenfunction ψ (r, θ, φ) R(r)Y lml (θ, φ) which involves the spherical harmonics Y lml that we briefly encountered in 3D rigid rotation. Replacing the above θ and φ terms with ˆL 2 and dividing through by 2mr 2 the original Schrödinger equation can be simplified h2 2m r 2 r r2 r R(r)Y lm l (θ, φ) + ˆL 2 2mr 2 R(r)Y lm l (θ,φ ) Ze 2 R(r)Y lml (θ, φ) ER(r)Y lml (θ, φ) r /9/7 But the Y lml are the eigenfunctions of ˆL 2 and we can replace the operation by its eigenvalue, ˆL 2 Y lml l(l + )h 2 Y lml so that h2 _ 2m r 2 r r2 r R(r)Y l(l + )h2 lm l (θ, φ) + 2mr 2 R(r)Y lml (θ,φ ) Ze 2 R(r)Y r lml (θ, φ) ER(r)Y lml (θ, φ)
2 -2- This last equation has no operations in θ or φ and the Y lml can be cancelled out of both sides of the equation leaving a one-dimensional radial equation in r to be solved 2m r 2 h2 d dr r2 d dr l(l + ) r 2 Ze2 r R nl (r) E n R nl (r) where we see that the radial function R nl (r) depends upon two quantum numbers. Eigenvalues, Quantum Numbers, and Nodes (5.2, 5.3) Energy Eigenvalues. The hydrogen atom Schrödinger equation is an eigenfunction/eigenvalue equation where the eigenvalues are the energies of the atom. Our text gives the energy eigenvalues as where a is the Bohr radius. E n a Z 2 me 4 8ε 2 h2 n 2 Z 2 e 2 8πε a n 2 Z 2 me 4 2n 2 h 2 Z 2 e 2 2n 2 a ε h 2 h 2 π me 2 me 2 (5.3,5.4) The left hand columns give the values in SI units. The right hand column may be easier to use in formulas. The right hand column values can be coverted into the left hand values if each e is substituted by e/ 4πε. The collection of constants e 2 /8πε a is known as the Rydberg constant whose value is J. The simplest expression for the energy of a hydrogenic atom is given interms of the Rydberg constant E n Z 2 n 2 Ry Quantum Numbers. There are three internal degrees of freedom for the hydrogen atom (excluding translation of the center of mass). Consequently, the solution to the Schrödinger equation gives three quantum numbers originating from the boundary conditions placed upon the solution of the differential equations. We will not immediately consider the spin quantum number, m 2. Itisonly derivable from a relativistic treatment (first accomplished by Dirac). Hydrogenic Quantum Numbers (5.5) quantum number symbol value principle n, 2, 3,... angular momentum l,,..., n magnetic m l, ±,..., ±l a 5.22 x - m geometric interpretation volume in space occupied by e - shape of the volume orientation of the volume Nodes. The above quantum numbers also allow one to determine the number and type of nodes in the wavefunction. Remember that a node is where the wavefunction passes through zero. Just as for the particle in a box and the harmonic oscillator, the total number of nodes is one less than the principle quantum number. Nodes are of two types reflecting the radial and angular nature of the wavefunction. The wavefunction for the lowest energy ground state does not have any nodes Nodes (end of Example Problem 5.2) type number total number of nodes n number of angular nodes l number of radial nodes n l
3 -3- Eigenfunctions ( ) In spherical polar coordinates text FIG 3.7 x r sin θ cos φ y r sin θ sin φ z r cos θ we must change the usual Cartesian integration dxdydz r 2 dr sinθ dθ dφ Normalization. Since ψ nlml R nl (r)y lml (θ,φ ) then N 2 ψ * nlm l ψ nlml d 3 τ > π 2π N 2 Y lm 2 l (θ, φ)sin θ dθ dφ r2 R 2 nl(r)dr Normalized Radial Functions R nl (r) where Z is the charge on the nucleus s R (r) 2 Z a 3/2 e Zr/a 2s R 2 (r) 3/2 Z 8 a 2 Zr a e Zr/2a 2p R 2 (r) 24 Z Zr a 3/2 a e Zr/2a Contour Plot of ψ 3s R 3 (r) 3p R 3 (r) 3/2 2 Z 8Zr Z 2 r a a 4 Z 6Zr 8 6 a 3/2 Z 2 r 2 a a 2 a 2 e Zr/3a e Zr/3a 3D Perspective Plot of ψ 3d R 32 (r) 4 Z Z 8 3 a 3/2 2 r 2 a 2 e Zr/3a 3D Perspective Plots of ψ 2, ψ 2 2,a ψ 2 3, 3D Perspective Plot of ψ 2 MOST PROBABLE LOCATION OF AN s ORBITAL IS AT THE NUCLEUS! r 2 R 2 r 2 R 2 r 2 R 2 3D Perspective Plot of ψ 3
4 -4- Two dimensional radial (spherical) nodes s Orbital radial nodes (l, ml > orbital) Two dimensional angular nodes Drums use two-dimensional surfaces (membranes) to produce sound. Drum heads vibrate in modes that are describable by radial nodal lines and another by angular nodal lines. The above radial nodal patterns can be compared with the nodes of hydrogen s orbitals while the angular node patterns on the right can be compared to hydrogen p and d orbitals. p Orbital angular nodes (l, ml -,, > 3 orbitals) AN ORBITAL IS A HYDRO- GENIC WAVEFUNCTION WITH SPECIFIED VALUES OF n, l, AND m l d Orbital angular nodes (l2, m l -2, -,, -,,2 > 5 orbitals)
5 -5- Probability Density, Probability, and Radial Distribution Function ψ nlml (r,θ,φ ) 2 is the probability density for finding the electron at the point (r,θ,φ ). The probability is given by the product of the probability density and the volume over which the probability is being determined. ψ nlml (r,θ,φ ) 2 r 2 dr sinθ dθ dφ is the probability for finding the electron "within the vicinity of a single point (r,θ,φ )", i.e., in the region of space where its coordinates lie in the range r to dr, θ to dθ, and φ to dφ. Due to the r 2 dr factor this probability is on a spherical shell of thickness dr but is only a small slice of it as shown in text Figure 3.7 given on p.3 of these notes. The probability for finding the electron at a given value of r irrespective of the angles is given byadding up the infinitesimal probabilities given above for all possible values of θ and φ while keeping r fixed. This probability for finding the electron in a thin spherical shell of constant radius which is centered at the origin with inner radius r and outer radius r + dr is found by integrating the above probability over θ and φ π 2π ψ nlm l (r, θ, φ) 2 r 2 dr sin θ dθ dφ R2 nl(r)r 2 dr π 2π where R 2 nl(r)r 2 is the radial distribution function, a probability density. Y lm l (θ, φ) 2 sin θ dθ dφ R2 nl(r)r 2 dr (5.9) r/a Probability density and radial distribution function for hydrogen s orbital. r 2 R 2 s(r) ψ s (r) 2 r 2 R 2 (r) s 2p 2s Radial distribution functions for hydrogen showing position of radial nodes in units of the Bohr radius, a. Note that the maximum in the s radial distribution function is exactly what Bohr predicted. As the principle quantum number increases the average separation r of the electron from the nucleus increases and r 2 (a measure of the width of the distribution) increases. r/a For fixed n, the average radius of the electron decreases with increasing l. A constant n means that the energy is constant. If l were to increase this implies that the kinetic energy is increasing. The only way to maintain the energy constant is for the potential energy to decrease. Since the potential is negative, it decreases (becomes more negative) as the radius decreases. So the 3d orbitals are closer to the nucleus on average than the 3p. r 2 R 2 (r) 3d 3p 3s r/a
6 -6Normalized Spherical Harmonics Ylml (θ,φ) Θlml (θ)φml (φ) Y 4π /2 /2 Y2 5 3 cos2 θ 6π Y3 7 6π /2 cos θ 5 cos2 θ 3 blue border figures: red + blue black background figures: green + red Y s Y, pz Y, dz2 Y2, fz3 Y3 3 4π /2 cos θ Y 2± 5 + 8π /2 cosθ sinθ e±iφ d xz Y ± 3 + 8π (Y 2 + Y 2 ) /2 ±iφ sinθ e Y2±2 d yz i π (Y + Y ) /2 sin2 θ e±2iφ px (Y 2 Y 2 ) py i (Y Y ) d x 2 y2 (Y 22 + Y 2 2 ) d xy i (Y 22 Y 2 2 ) Real Linear Combination Red and green images on the left of each panel are the traditional orbitals of chemistry. The images to the right in blue boxes are the actual spherical harmonics. As they are complex whenever ml >, Euler's formula is used to take a sum (real part) and difference (imaginary part) of the +ml and ml wavefunctions with the same l. The square then gives the chemistry orbitals. iφ e e iφ eiφ +/ e-iφ cos φ + i sin φ cos φ (eiφ + e iφ )/2 add iφ iφ cos φ i sin φ subtract sin φ (e e )/2i x r sin θ cos φ y r sin θ sin φ z r cos θ Y2, 2 ~ Y2, + Y2,- 2 ~ Y2, Y2,- 2 p z cos θ p x sin θ cos φ p y sin θ sin φ d z2 3 cos2 θ d xz cos θ sin θ cos φ d yz cos θ sin θ sin φ dxy sin2θ sin 2 φ dx2 y2 sin2θ cos 2 φ
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