Physical Properties of Molecular Materials

Physical Properties of Molecular Materials

Vibrational States

Introduction

H(R_i, B_A | E_e(R)) = E_g P_q (r, R) is associated to the full set of quantum numbers that define a particular possible state of the system:

• {r_i} = 3N_e / | 3 | N_e electrons
• {R_i}
• {r_A} = 3N_n / | R_i | N_n nuclei

H(R, B) = T_e(r) + T_N(R) + V_ee(r) + V_eN.

V_ee = V_ee + V_i V_mm. V_eN= - ∑_i ∑_R e² Z_i. / | R_i |

By Coulomb interaction:

• one-body operator. non-physical potential self-interactions are excluded and each interaction pair is considered only once.
• They are two-body operators but electron-nuclease attraction term comes out in the form of a one-body operator when it is considered from the point of view of the electrons.

Born-Oppenheimer Approximation

The electronic state can be described within a quasistatic field generated by almost immobile nuclei (nuclei are less freely participative). For a given quantum state | e, E > one could introduce both the electronic wave functione_σ as a parametric function with respect to the positions of fraction of the nucle, (E) and the nuclear wave function φ_n(R).

H(> (R, B), = Φ_e ( | B) φ_n (R)

H( ( R, B), ) E Hen(R, B) = E^g φ_n (R) E: total energy of the molecular quantum state, including nuclear and electronic contributions.

T_N + _B V_mm (r)

::..

By multiplying form the left by Φ^⊆ and integrating over V _eN 걸=H_ee(R)

H_e = T_e + V_eN (r) + V_ee | Ψ_{n e . He = Eg Ψe, electronic Hamiltonian (dependence upon B)}

Scharedinger eq for the nucle:

_n

_n

H_ee (R)

It contains V_eN + parametric with respect to R - both a general/volume (electronic energies) and a generates _{e electronic wave function as parametric (UNI)}

The wave function of the nuclei Δ is a function of an effective nuclear Hamiltonian dependent:

adoptive approximation: V_{eg/cm one neglection doesn't affect E}

to both stored the nuclear nuclei static and nuclease these terms carries a | numerical part an the neglect of the electronic is translated as this kind requires

Hellmann-Feynman Theorem

The nuclite are to be fixed in position as point charges and the force required to be applied to them λ is to be calculated. Let λ be one of any number of parameters which specifies nuclear position. Index i, it is to be associated with λ in a such way that λdλ measures the virtual force.

Eigenstates of the Quantum Harmonic Oscillator

_H |φ_k(q)> = E_k |φ_k(q)> The integer quantum number k labels the different vibrational eigenstates _H|v|k|> = E|k|k

Dimensionless Hamiltonian ℏ_H(q) = ¹⁄₂( _p^² + 1 + 4^{q^2 2)}

Dimensionless momentum operator _p^ = -iℏ_∂⁄_∂q q^(t) = ¹⁄₂ (_p^² + t²)

We know that [x, p_x] = iℏ, [t, p] = -iℏt => t[p, βt]

=> t²[p², βt] = tc, _a^^ = ⊂ ^⊂t - i_pt i_p ¹⁄₂ &sub, . . .

Therefore N = ¹⁄₂ (Mean a_^† = A^+theta+4 + . . .

¹⁄₂

1) ^{[a^^†} ≥ 0

2) The ⊂ operator increases by one the eigenvalue

N^U = _a^^† _N^U + 1 _];) a^

> |n> = a_^†_ÃU^ ^^p |n> = (n+1)_Ã

3) The à operator decreases by one the eigenvalue

[N^]= (_a^^†⊃); _^2V; |a^^e-3&asymp

a^t - sub + [sub ^N>N^{N^-4]}

= (n-1)_a&j

Eigenfunctions

The ground state is such that the action of ⊂ on it gives zero Lomega: b_Ãe⁄

The Gaussian function Φ₀=exp(-t²⁄2) satisfied this differential equation.

Φ₀(t=0) = ∫_E&f; exp(cl^-t²)

=>&nin º

¹0;&E;

. . .

(n+¹⁄_2⁄aL⁄h ⁄⊂ná - n _†)

¹0;SOL _{"n ⁿ+½}^,q,

. . .

Energies En = h _&ots&lcal = ¹1; _{(u+n, e+, ^e⁄/e⁄}·

^⁄

ELECTRONIC STRUCTURE OF DIATOMIC AND POLYATOMIC MOLECULES

The structure of molecules of the macroscopic level depends upon the quantum mechanics among the constituent atoms. The 3D structure is generally due to the way valence level form and occupy 3D themselves.

Valence electrons belonging to atoms in complete shells, once they are tightly bound to the core (there are electrons in the general context that one is more green). They are slightly perturbed by the bonding mechanism.

• Bonding MO: σ_g = (1s_A + 1s_B)
• Antibonding MO: σ_u* = (1s_A - 1s_B) antibonding MO (node/plane where it is needed, the bond)
• σ_u* antibonding MO

If antibonding contributions overshade bonding contributions → no bonding.

• π_u* bonding MO
• π_g bonding MO
• Pa x Py form π orbitals

Electrons in diatomic molecules feel an effective potential with cylindrical symmetry because of the net force acting on the excite (along the bond axis). Hence, the two components are conserved.

Quantum mechanically, this corresponds to the quantization of the corresponding expectation values: L_n (L_Z) = 0, +1

From O₂ to N₂: σ_u(π_g) atomic potential energy of O₂ antibond. This phenomenon is possible because 2s + 2p mixing rises energy of O₂ antibond. This phenomenon is possible because 2s + 2p orbital degeneracy exists.

Bond Order: BO = (n - n*) / 2 gives an approximate idea of the presence of the chemical bond.

Magnetism is associated with unparied electrons independent of conditions (O₂, α_gold).

E_eau

R O 2H bond breaking → the unsaturated KS solution irs inaccurate.

Preference to stay on the H side (asymmetric placed electron density. The antibonding MO is centered on Li side instead.

Interaction between level which are close.