7/30/2019 Physics Formula 60

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52 Physics Formulary by ir. J.C.A. Wevers

For N particles the symmetric spatial function is given by:

S(1,...,N) =

(all permutations of1..N)

The antisymmetric wavefunction is given by the determinant A(1,...,N) =1N!

|uEi(j)|

10.15.2 Molecules

The wavefunctions of atom a and b are a andb. If the 2 atoms approach each other there are two possibilities:the total wavefunction approaches the bonding function with lower total energy B =

12

2(a + b) or

approaches the anti-bonding function with higher energy AB =12

2(a b). If a molecular-orbital is

symmetric w.r.t. the connecting axis, like a combination of two s-orbitals it is called a -orbital, otherwise a-orbital, like the combination of two p-orbitals along two axes.

The energy of a system is: E=|H|| .

The energy calculated with this method is always higherthan the real energy if is only an approximation forthe solutions ofH = E. Also, if there are more functions to be chosen, the function which gives the lowestenergy is the best approximation. Applying this to the function =

cii one finds: (Hij ESij)ci = 0.

This equation has only solutions if the secular determinant|Hij ESij | = 0. Here, Hij = i|H|j andSij = i|j. i := Hii is the Coulomb integral and ij := Hij the exchange integral. Sii = 1 and Sij isthe overlap integral.

The first approximation in the molecular-orbital theory is to place both electrons of a chemical bond in the

bonding orbital: (1, 2) = B(1)B(2). This results in a large electron density between the nuclei andtherefore a repulsion. A better approximation is: (1, 2) = C1B(1)B(2)+C2AB(1)AB(2), with C1 = 1and C2 0.6.In some atoms, such as C, it is energetical more suitable to form orbitals which are a linear combination of the

s, p and d states. There are three ways of hybridization in C:

1. SP-hybridization: sp = 122(2s 2pz ). There are 2 hybrid orbitals which are placed on one lineunder 180. Further the 2px and 2py orbitals remain.

2. SP2 hybridization: sp2 = 2s/

3 + c12pz + c22py , where (c1, c2) {(

2/3, 0), (1/6, 1/2), (1/6,1/2)}. The 3 SP2 orbitals lay in one plane, with symmetry axes which are at an angle of120.

3. SP3 hybridization: sp3 =12(2s 2pz 2py 2px). The 4 SP3 orbitals form a tetraheder with the

symmetry axes at an angle of10928.

10.16 Quantum statistics

If a system exists in a state in which one has not the disposal of the maximal amount of information about thesystem, it can be described by a density matrix . If the probability that the system is in state i is given by ai,one can write for the expectation value a ofA: a =

i

rii|A|i.

If is expanded into an orthonormal basis {k} as: (i) =k

c(i)k k, holds:

A =

k

(A)kk = Tr(A)

where lk = c

kcl. is hermitian, with Tr() = 1. Further holds =

ri|ii|. The probability to findeigenvalue an when measuring A is given by nn if one uses a basis of eigenvectors ofA for {k}. For thetime-dependence holds (in the Schrodinger image operators are not explicitly time-dependent):

ih

d

dt = [H, ]