Quantum Mechanics Flashcards
wavefunction
the state of a system is fully described by a mathematical function Ψ,
Probability interpretation of Ψ (Born Interpretation)
consider a particle which can move only in the x direction
this particle is described by a wavefunction Ψ(x) - wavefunction only depends on x
assume that the wavefunction is normalized
the probability of finding the
particle somewhere along the x direction is 1
the wavefunction itself has no physical meaning
- it may, at any given point in space, be positive or negative, real or complex
observable
a measurable property such as bond length, dipole moment, kinetic energy
every observable B is represented by an operator
- all operators can be built from the operators for position and momentum.
operator
the Hamiltonian
The role of the operator is to operate on a wavefunction to yield information associated with the observable that the operator represents.
eigenvalue equation
The Schrödinger equation - for a particle moving in the x direction
an exact wavefunction
the eigenvalue E is an energy
orthogonal wavefunctions
Any two non-degenerate solutions (i.e. solutions of different energy) are orthogonal
orthonormal wavefunctions
expectation values
The expectation value of an operator B for a wavefunction Ψ is denoted < B > and is defined as:
- dτ tells us that the integration is being performed over all space.
when a system is described by a wavefunction Ψ, the average value of the observable B in a series of measurements is equal to the expectation value of the corresponding operator Bˆ
when Ψ is an eigenfunction of Bˆ, determination of B always yields one result, b.
When Ψ is not an eigenfunction of Bˆ, a single measurement of B yields a single outcome which is one of the eigenvalues of Bˆ, and a large number of measurements will yield an average of the eigenvalues of Bˆ
the variation principle
for any trial wavefunction Ψtrial , the expectation value of the energy can never
be less than the true ground state energy E0
The expectation value of the energy is an average of the true energies of the system E0 , E1 , E2 …., and this can never be less than E0
harmonic oscillator
harmonic oscillations occur when a system experiences a restoring force proportional to the displacement from equilibrium, e.g. pendulums, vibrating springs.
consider a one-dimensional harmonic oscillator:
k , the constant of proportionality, is called the force constant.
solution for the harmonic oscillator
- determine value of α required for wavefunction Ψ0 to be eigenfunctions of Hamiltonion
we seek a wavefunction such that the total energy E is constant as it must be an eigenvalue to be an exact solution.
solution for the harmonic oscillator
- determine the eigenvalues of H operator acting on Ψ0
wavefunctions associated with harmonic oscillator
the energies have even spacing, ΔE = ℏω
- If the force constant k increases, ω increases and hence so does ΔE
- This will also be the case if the mass decreases
zero-point energy
the eigenvalue of the ground state E0 ≠ 0.
harmonic oscillator - applications in spectroscopy
potential between two atoms in a diatomic molecule can be represented schematically by the plot of V(x) against x
well is very steep for small x due to the large replusion between the nuclei
potential tends to zero at large x as the bond weakens then breaks
harmonic oscillator potential, shown by V^(2)(x) on the plot, is a good approximation to V(x) around the
equilibrium (most stable) internuclear separation xe, but clearly not away from the equilibrium region
Morse potential
simple harmonic oscillator is used as a first approximation, and is most accurate for the ground state energy (v = 0), the zero-point energy
Morse potential more realistic for diatomic molecules compared to simple harmonic oscillator
term containing χe in Ev equation: reduces energy from the harmonic oscillator value and becomes increasingly important as v becomes large
Thus the vibrational energy level separation is not constant as for the harmonic oscillator, but converges as v gets larger. The molecule dissociates as v →∞
particle on a ring
a particle of mass, m moving at a constant velocity, v around a circle of radius, r in the xy plane
The potential energy is constant - the Hamiltonian for this motion is just the kinetic energy part:
the particle is confined to move at a fixed radius - therefore use polar coordinates.
r is a constant, any term containing a derivative of r will vanish
the Shrodinger equation for particle on a ring - substituting the moment of inertia, I of the particle about the z axis
Normalisation constant for particle on a ring
N^2(2π) = 1
N = 1/ √2π
angular momentum for particle on a ring
The z component of the angular momentum of a particle moving at constant velocity in a circle, Lz , can have only certain values which are multiples of ℏ
The angular momentum is quantized.
Particle on a sphere – rotation in 3 dimensions
particle of mass, m which is free to move anywhere on the surface of a sphere, at a fixed distance, r from an origin
the potential energy is constant and can be neglected
the Shrodinger equation for particle on a sphere
solutions for particle on a sphere
energy is independent of ml and hence for every value of l there are (2l+1) states with the same energy (degenerate)
the first nine spherical harmonics - ignoring their normalisation constants
degenerate wavefunctions
any linear combination of them will have the exact same energy, and be a solution to the corresponding Schrödinger equation.
rigid rotor – the rotation of diatomic molecules
consider a diatomic molecule AB
- The atoms rotate about the centre of mass
- the motion of the masses about the centre of mass is mathematically equivalent to the rotation of a single particle of reduced mass about a fixed point
distance between the fixed point and the particle is equal to the bond length r
- treat the rotation of a diatomic molecule as if it were the motion of a particle of mass μ on the surface of a sphere
energies of the wavefunction for particles on a sphere
In rotational spectroscopy we normally use J as the symbol for the quantum number
the hydrogen atom
consider general case of hydrogenic atoms - so solutions can be extended to He+ and Li2+
Coulomb’s law: gives the potential energy of the electron in the electrostatic
field generated by the nucleus
atomic units
QM generally uses atomic units
in SI units a0 = 5.29177208x10-11 m.
The Hamiltonian for H atom
the potential V is independent of the angles θ and φ
the part of the Hamiltonian that depends on these angles will be the same as for the rigid rotor.
The kinetic energy operator only differs in that r can now vary (r is no longer constant)
The Schrodinger equation for H atom
expression for the energy is in atomic units
- atomic unit of energy is also called a hartree,
(1 Eh = 4.3597438 x 10-18 J = 27.211383 eV).
the energies depend only on the quantum number n
- This is true only for the H atom, or any other atomic ion with only one electron
The Hamiltonian for the He atom
assume that the repulsion between the two electrons can be mathematically ignored and somehow absorbed into an effective potential (ignore 1/r12 term)
are thus assuming that the electrons move independently of one another
- assign each electron its own hydrogenic wavefunction
wavefunction for He
electron spin
Pauli exclusion principle
No two electrons in an atom may have the same set of the four quantum numbers n, l, ml and ms
consider the effect on Ψ(1,2) of interchanging the two electrons,
i.e. Ψ(1,2) → Ψ(2,1)
electrons are indistinguishable and hence this process cannot affect the physical properties of the system
the probability distribution Ψ*Ψ must remain unchanged.
- For this to be true: either Ψ(1,2) = Ψ(2,1) or Ψ(1,2) = -Ψ(2,1)
the second condition is the correct restriction for electrons - therefore the more fundamental form of the Pauli exclusion principle:
the total wavefunction of a system must change sign when any two electrons are interchanged
- the total wavefunction must be antisymmetric
the ground state of the He atoms
α - spin up
β - spin down
electrons are indistinguishable
- cannot say for sure that electron 1 has α spin and electron 2 has β spin (or vice versa)
when electrons have opposite spins, there must be equal probabilities of α(1)β(2) and α(2)β(1)
- which we can ensure by taking linear combinations of these spin arrangements
combine spin wavefunctions with spatial wavefunction for He atom
The spatial wavefunction is symmetric with respect to the interchange of the two electrons, and hence the spin wavefunction must be antisymmetric. Only the fourth spin function is antisymmetric
fourth wavefunction satisfies Pauli exclusion principle
Slater’s determinant - the ground state of the He atoms - written in determinantal form
Each term in the determinant has a hydrogenic spatial orbital multiplied by a spin function, and is known as a spin-orbital
excited states of the He atom
He electron configuration: 1s2
Excited states of He: promote electron from the 1s orbital to the 2s to yield the configuration 1s12s1
energy of singlet and triplet states of He atom
there is a difference in energy between the singlet and triplet states of 2K where the exchange integral, K > 0, and hence the triplet state is the more stable
Born-Oppenheimer approximation
the total wavefunction for a molecule can be approximated as: Ψmol = Ψnuc Ψelec
where Ψelec is determined from an electronic Hamiltonian (in atomic units)
i and j refer to the electrons, k and l to the nuclei
Thus, r_ik is the distance from electron i to nucleus k with charge Z_k
R_kl is the fixed internuclear distance.
Linear Combination of Atomic Orbitals (LCAO)
Linear Combinations of Molecular Orbitals
same approach for LCAO is taken for molecular wavefunctions.
the electronic wavefunction for a molecule with n electrons is written as the product of one-electron wavefunctions:
this is not actually a product wavefunction, but the leading diagonal of a Slater determinant
there will be as many molecular orbitals Ψi as there are atomic orbitals φk in the basis set.
secular equations
consider a diatomic molecule with
- atomic orbital φ1 on atom 1
- atomic orbital φ2 on atom 2
separated by a fixed distance R
Consider also a trial wavefunction for the molecular orbitals (MOs): Ψtrial = c1φ1 + c2φ2
HOW DO WE DETERMINE c1 AND c2
secular equations in matrix form
secular equations have non-trivial solutions only when the secular determinant equals zero:
Two-orbital systems: Identical orbitals and zero overlap
φ1 and φ2 are identical
- the combination of any two identical atomic orbitals on adjacent atoms of the same type (e.g. in homonuclear diatomics or the π orbitals of ethene, where φ1 and φ2 are each 2pz orbitals on the C atoms)
φ1 and φ2 are assumed real - we can therefore replace the Coulomb resonance integrals with α and β respectively
molecular orbital diagram formed from two identical atomic orbitals when we assume that there is no overlap i.e. S_12 = 0
the bonding orbital is stabilised by as much as the antibonding orbital is destabilised.
Coulomb integral, α
- the energy an electron would have in a molecule if it occupied either AO φ1 or φ2
- more negative then the energy electron would have if it occupied φ1 or φ2 in the isolated atoms - due to Coulomb attraction between electron and second nucleus
Resonance Integral,
- energy of interaction between φ1 and φ2 in the molecule
- normally negative
- typically about five times smaller than α for adjacent, identical orbitals
normalise coefficients
Two-orbital systems: Identical orbitals and non-zero overlap
anti-bonding orbital is destabilised by
more than the bonding orbital is stabilised
He2 is not stable
- two electrons in both the bonding and anti-bonding
orbital
- giving an energy of 2E+ + 2E- which is less
stable than the isolated atoms at all separations.
Two-orbital systems: Different orbitals and zero overlap
in heteronuclear diatomics, φ1 and φ2 are not the same
Two-orbital systems: Different orbitals and zero overlap - when φ1 and φ2 have very different energies
α1 and α2 will be very different from one another and δ^2»_space; β^2
atomic orbitals will mix together strongly to form molecular orbitals only if
If two atomic orbitals have very different energies (and hence α1 and α2 are very different) then
they will not mix together strongly.
Huckel Theory
In conjugated organic molecules such as ethene or benzene, the π orbitals usually have higher (i.e. less negative) energies than the σ orbitals
- are the most important in chemical reactions and spectroscopy.
Huckel theory: an approach to the electronic structure of conjugated molecules
- ignores the σ orbitals
(only using them to define the molecular geometry)
- determines the π MOs for systems with n conjugated carbon atoms
general approach to Huckel theory
approximations in Huckel theory
Huckel theory - secular equations written in matrix notation
Huckel theory - setting up the matrix
α - E along the leading diagonal
β along the off-diagonals when atom j is next to atom k
zero along the off-diagonals when atom j is not next to atom k
Huckel theory - coefficients are normalised
Huckel theory - total energy of molecule
There are n electrons for an uncharged molecule with n C atoms contributing to the π system and these can now be put in pairs into the MOs.
ni (the occupation number of the ith MO) takes the values 0, 1 or 2 (it is the number of electrons)
Huckel theory example - the allyl radical
there are three 2p orbitals, one on each carbon atom, that can contribute to the π MOs
- π MOs are perpendicular to the molecular plane
there are three atomic orbitals therefore, there must be three π MOs
secular determinant must equal zero for non-trivial solutions
Huckel theory example - the allyl radical: for each E value, the MO coefficients are found by substituting E back into the secular equations
Huckel theory example - the allyl radical: MO diagram
electron population on atom k (the π electron density)
since each atom k has a residual nuclear charge of +1. These sort of charge data are useful for determining the most likely position in a molecule for attack by an electrophile or nucleophile
π bond order between atoms j and k
