Quantum Chemistry I : CHEM 565

Quantum Chemistry I : CHEM 565 Lasse Jensen October 26, 2008 1

1 Introduction This set of lecture note is for the course Quantum Chemistry I (CHEM 565) taught Fall 2008. The notes are at this stage rather incomplete and should be checked for errors. Therefore, the notes are not to be distributed in any form without the accept from the author. L. Jensen, 2008 2 Background Before starting discussion the inner workings of quantum mechanism and how to use it to describe chemistry, i.e. the topic quantum chemistry, it is worth to discuss briefly a few of the realizations that lead to the development of quantum mechanics. 2.1 Black-body radiation One of the fundamental problems in physics at the end of the nineteenth century was to explain the black-body radiation. A black-body is an object that absorbs all light that falls on it, and thus appear black when cold. In the laboratory, black-body radiation is approximated by the radiation from a small hole entrance to a large cavity. Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped, in which process it is nearly certain to be absorbed. This occurs regardless of the wavelength of the radiation entering (as long as it is small compared to the hole). The hole, then, is a close approximation of a theoretical black body and, if the cavity is heated, the spectrum of the hole s radiation (i.e., the amount of light emitted from the hole at each wavelength) will be continuous, and will not depend on the material in the cavity (compare with emission 2

spectrum). Two characteristics of the radiation had been identified at the end of the century. The first is expressed in the Stefan-Boltmann law: J = σt 4 (1) which states that the total energy radiated per unit surface area, J of a black body in unit time is directly proportional to the fourth power of the black body s temperature T, where the constant is independent of material. The second law is Wien s displacement law that states that there is an inverse relationship between the wavelength of the peak of the emission of a black body and its temperature λ max = b (2) T The radiated energy can be considered to be produced by standing wave or resonant modes of the cavity which is radiating. The amount of radiation emitted in a given frequency range should be proportional to the number of modes in that range. The best of classical physics suggested that all modes had an equal chance of being produced, and that the number of modes went up proportional to the square of the frequency. Based on classical physics RayleighJeans proposed the following law to explain the spectra which relates the energy per unit volume per unit wavelength as u λ (T )= 8πk BT λ 4 (3) which posed the famous ultraviolet catastrophe, where for small wavelength the formula diverges. The correct relation was given by Planck five years earlier as u λ (T )= 8πhc 1 (4) λ 5 exp(hν/kt 1) In constructing a derivation of this law, he considered the possible ways of distributing electromagnetic energy over the different modes of charged oscillators in matter. Planck s law emerged when he assumed that the energy of these oscillators was limited to a set of discrete, integer multiples of a fundamental unit of energy, E, proportional to the oscillation frequency ν: E = hν (5) 3

2.2 The photoelectric effect The photoelectric effect is that when a metallic surface is exposed to light electrons are emitted. The observations from experiments are No electrons are emitted below a material dependent cutoff frequency. The number of electrons emitted are directly proportional to the intensity of light. The kinetic energy of the emitted electrons increase with increased frequency. Electrons are emitted instantly even for low intensities. This is at odds with classical wave theory of light, which predicted that the energy would be proportional to the intensity of the radiation. Einstein showed that this could be resolved by describing light as composed of discrete quanta, now called photons, each having an energy, E = hν. Conservation of energy then gives hν = Φ + T (6) where Φ is the work function of the metal, i.e. minimum energy needed for an electron to be emitted, and T the kinetic energy of the electron. 2.3 The Bohr model Bohr introduced his model for the atom based on Rutherford s planetary model of the atom as an attempt to describe the Balmer series. The Balmer series is the empirical relation describing a series of spectral transitions in 4

hydrogen as states that the frequencies of light that can be emitted from an excited hydrogen atom is given by The assumptions in the Bohr model is 1 λ = R( 1 2 1 ),n=3, 4, 5, (7) 2 n2 Electrons exists in circular orbits with discrete quantized energies. The laws of classical mechanics do not apply when electrons jump from one allowed orbit to another. The energy difference between orbitals is given by a single quantum of light (called a photon). The allowed orbits depend on quantized (discrete) values of orbital angular momentum. In a circular orbit the centrifugal force balances the attractive force of the electron F cf = mv2 = F Cou = e M 2 (8) r r 2 wherem is the mass of the electron, v is the speed of the electron, r is the radius of the orbit and e is the charge on the electron or proton. The energy of the orbiting electron is E = T + V = 1 2 mv2 e M 2 r = 1 2 e M 2 r which follows from the centrifugal force expression. The angular momentum assumption of the Bohr model implies (9) L = mvr = n,n=1, 2, 3, (10) which implies that, when combined with the centrifugal force equation, the radius of the orbit is given by (rmv) 2 rm = e2 M (11) n 2 2 mr = e2 M (12) r = n2 2 me 2 M (13) 5

This implies, from the energy equation, E n = 1 2 e M 2 r = 1 2 ( mem 4 2 ) 1 n 2. (14) The difference between energy levels recovers the Balmer series. The Bohr assumptions recover the observed Balmer series. The Bohr assumptions themselves, however, are not based on any more general theory. Why, for instance, should the allowed orbits depend on the angular momentum? 2.4 de Broglie Hypothesis The de Broglie hypothesis provides some insight. We know from the work of Einstein that the energy of a photon is E p = hν (15) Although the photon has zero rest mass its mass due to its motion is given my E p = mc 2 (16) Therefore the momentum of the photon is p = mc = E p /c = hν/c = h/λ (17) which illustrates the wave-particle duality of photons. De Broglie was the first to generalize this by hypothesizing that this is true for all mater λ = h p (18) If we assume that the electron has a momentum given by the de Broglie hypothesis, then the angular momentum is given by ( ) 2π L = pr = r (19) λ where λ is the wavelength of the electron wave. If only standing electron waves are permitted in the atom then only orbits with perimeters equal to integral numbers of wavelengths are allowed: λ = 2πr n. (20) 6

This implies that allowed orbits have angular momentum L = n (21) which is Bohr s fourth assumption. This leads to the so-called wave-particle duality of matter. 2.5 The uncertainty principle The wave-particle duality has significant effects on out ability to simultaneous measure certain variables. Consider a beam of particles with momentum, p, traveling in the y-direction. If the particles are described by a plane wave passes through a narrow slit in a wall, like a water-wave passing through a narrow channel the particle will diffract, and its wave will come out in a range of angles. The narrower the slit, the wider the diffracted wave and the greater the uncertainty in momentum afterwards. The size of the slit, w, is then related to the uncertainty in the position of the particle, i.e x = w. Similarly, the diffraction due to the slight causes a spread in the momentum, which is given by its projection in the x-direction as p x = psinα, where α is the angel to the first diffraction minimum, i.e. the extra distance travel from the center most be 1/2λ. Thus, at the slit x p x = wpsinα (22) Using simple trigonometric argument we can relate the extra distance travel from the center to be wsinα. Therefore, we can write the uncertainty as where we used de Broglie relation. x p x pλ = h (23) 2.6 The time-dependent Schrödinger equation We know from classical physics that the total energy E of a particle is given by E = T + V = p2 2m + V (24) where m is the mass of the particle, and p the momentum. We know from Einstein that the energy E of a photon is proportional to the frequency ν of 7

the corresponding electromagnetic wave: E = hν = h (2πν) = ω (25) 2π where ω =2πν is the angular frequency of the wave. We also know from de Broglie hypothesis that any particle can be associated with a wave, represented mathematically by a wavefunction Ψ, and that the momentum p of the particle is related to the wavelength λ of the associated wave by: p = h λ = h 2π 2π λ = k (26) where k = 2π/λ is the wavenumber of the wave. Schrödinger s great insight, late in 1925, was to express the phase of a plane wave as a complex phase factor Ψ(r,t)=Ae i(k r ωt) (27) Differentiation of the wave function with respect to time gives Ψ = iωψ (28) t Using this we can establish a relationship between the time-derivative of the wave function and and the energy as EΨ = ωψ = i t Ψ (29) In a similar manner, if we differentiate the wave function with respect to the direction x, we obtain x Ψ = ik xψ (30) We can now establish a connection between the momentum of the particle and the spatial derivatives of the wave function as and similarly for the momentum squarred p x Ψ = k x Ψ = i x Ψ (31) p 2 xψ = 2 2 x 2 Ψ (32) 8

and if we generalize this to all three directions we get ( ) p 2 Ψ =(p 2 2 x + p2 y + p2 z )Ψ = 2 x + 2 2 y + 2 Ψ = 2 2 Ψ (33) 2 z 2 Inserting these expressions for the energy and momentum into the classical formula we started with we get EΨ = EΨ (34) i t Ψ = p2 2m Ψ + V Ψ (35) i t Ψ = 2 2m 2 Ψ + V Ψ (36) i Ψ = ĤΨ (37) t which is the time-dependent Schrödinger s equation. 2.7 The time-independent Schrödinger equation Most of the time we do need to consider the complete Schrödinger equation but rather a simpler form. Lets consider the situation where the potential is time-independent i Ψ(r, t) = t 2 ĤΨ(r, t) = 2m 2 Ψ(r, t)+v (r)ψ(r, t) (38) For now we will restrict to look at solutions of the form If we now do the partial derivatives we get and Ψ(r, t) =f(t)ψ(r) (39) Ψ(r, t) t = df (t) ψ(r) (40) dt Ψ(r, t) =f(t) ψ(r) (41) Substituting these into the time-dependent Schrödringer equation gives i df (t) 2 ψ(r) = f(t) ψ(r)+vf(t)ψ(r) (42) dt 2m 9

Now we divide by the wave function on both sides to get i 1 df (t) = 2 f(t) dt 2m 1 ψ(r)+v (43) ψ(r) Since the two sides of the equation depends on different parameters this equation can only be true if equal to a constant, which we will for now call E. Equating the left side of the equation to E we get Integrating these two equations we get 1 df (t) = f(t) Therefore, we get for the time-dependent function df (t) f(t) = E dt (44) i E dt (45) i lnf(t) = E i t + C (46) f(t) =e C e Et/i = Ae iet/hbar (47) which is what we started with. Equating the right hand side with E gives E = 2 1 ψ(r)+v (48) 2m ψ(r) Eψ(r) = 2 ψ(r)+vψ(r) 2m (49) Eψ(r) =Ĥψ(r) (50) which is the time-indendent Schrödinger equation, and where we postulate that the constant E gives the total energy of the system. Therefore, when the potential is time-indenpendent the total wave function is given by Ψ(r, t) =e iet/ ψ(r) (51) 2.8 Interpretation of the wave function The correct interpretation of the wave function for a particle was given by Born. He postulated that the probability at time t of finding the particle in some small region of space dr is given by Ψ(r, t) 2 dr (52) 10

Thus, the square of the wave function is the probability density for finding the particle. Therefore, we have Ψ(r, t) 2 dr = 1 (53) Lets consider this the probability density for our time-dependent wave function Ψ(r, t) 2 = Ψ(r, t) Ψ(r, t) (54) = [ e iet/ ψ(r) ] e iet/ ψ(r) (55) = e iet/ ψ(r) e iet/ ψ(r) (56) = ψ(r) ψ(r) (57) = ψ(r) 2 (58) and we see that the probability density is independent of time, and we call such states stationary. 11