Pchem - Section 3: Quantum Mechanics

Quantum Chemistry: History and Concepts, Simple Analytical Quantum Mechanical Model Systems, Modern Quantum Mechanical Problems: Atomic Systems, Symmetry, Molecular Orbital Theory, Spectral Properties, Advanced Topics: Electronic Structure Theory and Spectroscopy,

Because the nuclear motions are much slower than those of the electron, the molecular Schrodinger equation for the electron motion can be solved by assuming that the nuclei are at fixed locations. This is

the Born-Oppenheimer approximation

According to the Heisenberg Uncertainty Principle, if the operators for two physical properties do not commute then

the product of the two uncertainties is > or = h/4π

The requirement that wavefunctions for electrons in atoms and molecules be antisymmetric with respect to interchange of any pair of electrons is

pauli exclusion principle

For electrons emitted due to the photoelectric effect, the kinetic energy is a function of and the current is a function of of the incident light

For electrons emitted due to the photoelectric effect, the kinetic energy is a function of __**frequency**__ and the current is a function of __**intensity**__ of the incident light

Two wavefunctions, Ψ1 and Ψ2 are orthogonal if the integral of equals

integral of __psi 1* times psi 2 dT__ equals __0__

An eigenfunction of the operator d/dx is

e^ax

A sum of two degenerate eigenfunctions for an operator is still an eigenfunction for that operator

n/a

If Ψ(x) is the normalized wave function for a particle in one dimension, the average value for the momentum can be calculated using the momentum operator, (px^) from the integral

∫Ψ(x)\*(px^)Ψ(x)dx

integral of ( psi\* (px^) Psi ) dx

In a photoelectric effect experiment with photos of energy greater than the work function for the material, the number of photoelectrons ejected from a metallic surface depends on

the ***intensity*** of the light hitting the metal

Which function is an acceptable wave function over the indicated interval?

E^-x

range is 0 to infinity

The commutator of two operators, A and B , \[A,B\] equals 0. The physical properties *A* and *B* associated with the two operators.

may be simultaneously determined with unlimited precision.

In the Schrodinger equation the quantity, Hamiltonian represents the

total energy operator

If operators do not commute than they do not equal

zero

Franck-Condon Principle: idea that nuclei are and compared to electrons

because of this an electronic transition takes place in presence of fixed nuclei

slow moving and heavy

Expectation value

Integral of psi\*A^psi dT

Probability density function

Integral of psi\*(x) psi dT

De Broglie postulated that the wavelength of a panicle is inversely proportional to its momentum. The constant of proportionality is

h, plank’s constant

The quantum mechanical operator for velocity is given by

\-ih/m d/dx

There are many mathematically acceptable solutions for the Schrodinger equation for any particular system, but only certain ones are physically acceptable. This is because

physically acceptable wave functions must be finite, single valued, and continuous; and have a continuous first derivative.

A nodal surface is best described as a surface separating regions of sign for a wave function.

__**different**__

De Broglie did NOT use the…

schrodinger eqn

Harmonic oscillator: energy levels are evenly spaced, deltaU=+-1, and a changing dipole moment all come together so a line appears in the absorption spectrum

__**single**__

Particle in a box was created to understand spec containing alternating double and single bonds.

Uv-vis

Harmonic Oscillator: in V=1, there are two hills and only valley in the middle.

In V=2, valleys

In V=3, valleys

one, two, three

Adding cubic terms to the Hooke's law potential would transform it to an potential.

__**anharmonic**__

Anharmonic function example: Morse Potential

Why?

it involves an exponential function in the potential

A better potential energy function for the vibration is the Morse potential.

Solving the Schrodinger equation using the Morse potential gives energy levels that get as v increases leading to dissociation at the plateau region of the potential.

__**closer together**__

The Morse and harmonic oscillator potentials resemble each other closely at the of the energy well.

__**bottom**__

The Morse potential also has a zero point energy when v = ?

0

Ir needs a dipole moment greater than 0

vibrating atoms in a vibrational mode have a varying dipole moment during the vibration. This results in v = ± I.

For the rotation of molecules, the model is used. In this approximation the internuclear distance remains as the molecule rotates.

__**rigid rotor, constant**__

Allowed spectroscopic transitions require a value for the transition moment integral

non-zero

Which is a probability density function?

B

The energy of a particle in a three dimensional box with equal sides is given by E(n1,n2,n3) = (N1^2+N2^2+N3^2)(h^2/8mL^2) quantum numbers defining an independent state. The degeneracies of the first, second, and third energy levels are

1, 3, 3

For a particle in a one-dimensional box with finite walls at *x* = 0 and *x = L,* which one of the following statements is true?

The average value of the position of the particle is L / 2 for any allowed state.

The most important defect of the simple harmonic oscillator (SHO) approximation for the upper vibrational levels of a molecule is that

All levels are evenly spaced

For a harmonic oscillator, V(x)=1/2Kx^2 and the wave function is Psi0(r). If an anharmonic energy term is given as V(x)=-1/6ax^3 then the energy for an anharmonic oscillator is

1/2 hv + integral \[ Psi0\*(x)\[-1/6zx^3\]Psi0(x) \] dx

\
\
comes from hamiltonian for anharmonic : H=H0+H1 and H1=-1/6ax^3

\
SE is E= Int \[Psi0\*(x)(H0-1/6ax^3)Psi0(x) \] dT

\
which expands to E= E0 + Int \[Psi0\*(x)(H0-1/6ax^3)Psi0(x) \] dT and E0=1/2hv

In the quantum mechanical solution for the rigid rotor, the square of the angular momentum is given by L^2= J(J+1)(h/2pi)^3

If J=2, the possible values for the z component of the angular momentum are

Mj= 2, 1, 0, -1, -2

The quantum number *J* labels the rotational quantum state



The eigenvalues for the z-component of the angular momentum slates ranging from *+J* to *-J* are called m,.

Rigid rotor: If J=#, mj=

Mj= +-J

Energy for each energy (rotational) level

E= J(J+1)H^2/2I

Energy for a transition between states (rotational energy levels)… Delta E=

allowed transitions have delta J =

Delta E= hv = 2(J+1)H^2/2I

allowed transitions have delta J = plus or minus 1

The intensities and peak positions shown here were computed for one branch in the rotation- vibration spectrum of HCI molecule in the microwave region using the rigid rotor model at 298 K. When the temperature is increased to 1000 K we would expect the spectrum to change so that

the maximum intensity would occur for a peak at a higher wavenumber.

A portion of the ro-vibrational spectrum of HF at some temperature T is shown in the choices. The appearance of the spectrum at a high temperature could be

\

One branch of the infrared spectrum of a diatomic polar molecule looks like a set of nearly equally spaced lines where the lines first and then in intensity.

But, for **high temperatures** we see a decrease in the intensity

__***increase, decrease***__

__**should NOT**__

In the rotational spectrum of DCI, the rotational line spacings compared to those of HCI are approximately

Halved ; B=1/mu. mu HCl= 1 Mu DCl=2 so BHCl/DDCl = 1/2

If the box length for a particle in a one- dimensional box is doubled the energy will be of the original energy.

1/4

A particle confined in a one-dimensional box has a lowest energy of £,. If the mass of the particle is doubled, the lowest energy will be

E1/4

so if __**BOX LENGTH**__ or __**MASS**__ is doubled, the __**ENERGY**__ for PIB is multiplied by…

1/4

The one-dimensional quantum mechanical descriptions of the harmonic oscillator and the particle-in-a-box with infinite walls *differ* in that HO wave functions into classically forbidden regions

HO wave functions __**penetrate**__ into classically forbidden regions, PIB wave functions **do not**.

Which is **not** spectroscopic evidence for the anharmonicity of the potential energy as a function of internuclear distance

unequal spacing of the rotational fine structure of infrared absorption bands

Compare the force constants for the bond strength in HCI and DCI.

k(HCI) roughly equal to k(DCI)

Given the figure which describes a Morse potential for a diatomic molecule, identity and describe region 3

This is the attractive region of the interaction where the molecule is most stable.

For a diatomic molecule considered as a rigid rotor, when J=1, the magnitude of the angular momentum is

hcross(2)^1/2

Which system DOES NOT have a zero point energy?

Rigid rotor

Which systems DO have a zero point energy?

Hydrogen atom.

1D Harmonic oscillator

particle in 1D box

Transition moment integral determine the of absorption lines.

intensity

When J=0 and Ej=0 the molecules is not rotating and only and motion is possible

only __translational__ and __vibrational__ motion is possible

Of the five orbitals in the *d* subshell o f a hydrogen-like atom, only the angular wave function for the m=0 , dz^2 orbital is real. How are the other 4 orbitals generated? by taking of the imaginary functions

linear combinations

A radial node is a region where the WF is

zero

The number of nodes in a hydrogen-like orbital is less than the principal quantum number.

one

The one electron of hydrogen can have either up or down spin which is +1/2 or -1/2.

When applying a magnetic field, the **spin of the electron** can **align parallel** to the magnetic field or **anti-parallel** or in opposite direction to it.

So, this lead to energy states.

Without the magnetic field, the orbital has two possible spin slates, which are ordinarily degenerate, which would lead to energy state

two, one

A certain hydrogen-like alom (with only one electron) emits light with a frequency of 2.2\* 10" Hz associated with *n* = 2 → *n* = I transition.

What frequency light will be emitted for the *n =* 3 to *n = 2* transition? How will you solve?

V3→2/V2→1 = constant (1/n2^2 - 1/n1^2) / constant (1/n2^2 - 1/n1^2)

= const(1/9-1/4)/const(1/4-1/1)

(-5/36) / (-27/36)

= 5/27 approx 1/5

The square of the wave function, Psi1s^2, for the hydrogen 1s orbital has a max at r=

0

In the hydrogen atom, the energy of a state depend on the azimuthal quantum number or angular momentum

does not

Azimuthal quantum number is

ml

Hamiltonian for H atom: KE of - KE of - PE of and

KE of __**whole atom**__ - KE of __**nucleus**__ - PE of __**nucleus**__ and __**electrons**__

The splitting between the two lines in the yellow for the Na atom is due to

spin-orbit coupling.

What is the expectation value of r, (r), for the 1s orbital of hydrogen where the WF is given?

(hint: its a longer integral)

Triple integral of psi\* r psi sin(theta) dr d(theta) d(phi)

In the absence of a magnetic field, the order of the hydrogen atom energy levels is

E1s< E2s= E2p

Measurement of the z component of angular momentum for an electron in the 2px, state will yield

plus or minus 1h

Which graph would be best used lo explain why the average electron-nucleus distance for the 1s state of the H atom is 0.529 A?

r^2R^2 vs r

The average value of the angle *phi* for an electron in a H atom 1s orbital is

pi

The Na atom has 2 lines in the yellow region of the spectrum. These arise from transitions between which energy levels?

3s → 3p

What is the symmetry of BF3?

D3h

A molecule is in an initial slate characterized by B, symmetry. What is the symmetry of the final when an electric dipole transition in the z-direction is allowed?

A1

Symmetry group C3H6

D3h

The 3P2 state of the oxygen atom has the 1s2 s2 2p4 configuration. To which excited state is an electronic transition allowed in the electric dipole approximation?

1s2 2s2 2p3 3p1 ; 3P1

The planar methyl free radical, CH3, is in the D3h symmetry group. In this symmetry group, how many planes of symmetry are there?

4

The combinations of atomic orbitals used in constructing appropriate molecular orbitals for BeH2 must have the same as the molecule

__**symmetry**__

*Trans-*1,2-dibromoethene possesses which elements of symmetry? \n

c2, oh, i

The water molecule belongs to the C2, symmetry group which contains only

 a 180 degrees rotation axis, two reflection planes, and the identity operation

Wave functions and electronic slates of heteronuclear diatomic molecules are not labeled "ungerade" or "gerade" because these molecules have no

center of inversion

NH3 has the same symmetry group as

CCl3Br

In this configuration, ferrocene does *not* contain which symmetry element?

hint: it isnt in the word ferrocene

i

Which molecule has energy levels described by an asymmetric top?

HCCl3

How many normal modes are present for H2o2

6

Molecules of the point group D2h cannot be chiral. Which symmetry element rules out chiral molecules in this point group AND exhibit a c4 axis?

Oh

The number of nondegenerate irreducible representations for the C2v point group is

4

Using the simple homonuclear molecule MO energy level scheme for the ground state ofthe diatomic boron molecule, we would predict that

Br2- will have unpair electrons

as the number of nodes increases, so does the

__**energy**__

When an electron is removed from N2, the N-N bond lengthens; whereas when an electron is removed from O2, the O-O bond shortens. The most important factor involved in this difference between nitrogen and oxygen is that electron comes from an MO in oxygen and MO in nitrogen.

electron comes from an __**antibonding**__ MO in oxygen but from a __**bonding**__ MO in nitrogen,

Which of these diagrams would be expected to represent the *highest* energy molecular orbital?

the one that is alternating dark light dark light on top/bottom

Which expression is the best choice for the spatial part of a molecular orbital wave function for the ground state for H2?

1sa(1)1sb(2) + 1sb(1)1sa(2)

molecular orbital wave function does not have and should contain only terms

multiplication

covalent

Because the nuclear motions are much slower than those o f the electron, the molecular Schrodinger equation for the electron motion can be solved by assuming that the nuclei are at fixed locations. This is

the Born-Oppenheimer approximation.

the simple Huckel molecular orbital method has the secular determinant equal to a determinant where ‘s are diagonal and neighbor #’s are , everything else is

where __x’__s are diagonal and neighbor #’s are __1__ , everything else is __0__

According to molecular orbital theory, which molecule should be least stable and has a bond order of __**zero**__?

Be2

Bond order formula

bonding-antibonding/2

Find bond order given (1sog)2 (1sou\**)2 (2sog)2 (2sou**)2 (2pog)2 (2ppiu)4 (2ppig\*)3

Bonding e= 2+2+2+4 =10

antibonding e= \* = 2+2+3=7

BO = B-AB/2

BO=10-7/2 = 3/2