1. Introduction to basic algebra.
• Vectors.
• Matrices.
• Determinants.
• N-dimensional complex spaces (Dirac notation).
• Basis exchange.
MFQC - Alcamí 1
, Motivation: Vectors, Matrices and Quantum Chemistry
Real space: a vector in the plane can expressed as a linear combination of the vectors of the basis
a = a1e1 + a2 e2 = 4e1 + 3e2
e2 a ⎛ 4⎞ ⎛ 1⎞ ⎛ 0⎞
a = ⎜ ⎟ = 4 ⎜ ⎟ + 3⎜ ⎟
⎝ 3⎠ ⎝ 0⎠ ⎝ 1⎠
e1
, Motivation: Vectors, Matrices and Quantum Chemistry
Wavefunction. How express it as a vector?
Wavefunction: if we define a basis (i.e. pz orbitals on C
atoms), the benzene molecular orbitals can be expressed
as a linear combination of the vectors of the basis
{ }
Base (2 p1z ) + (2 pz2 ) + (2 pz3 ) + (2 pz4 ) + (2 pz5 ) + 2( pz6 )
ψ 1 = 1(2 p1z ) + 1(2 pz2 ) + 1(2 pz3 ) + 1(2 pz4 ) + 1(2 pz5 ) + 1(2 pz6 )
ψ 2 = 1(2 p1z ) + 1(2 pz2 ) + 0(2 pz3 ) − 1(2 pz4 ) − 1(2 pz5 ) + 0(2 pz6 )
MFQC - Alcamí 3
, Motivation: Vectors, Matrices and Quantum Chemistry
Real space: a vector in the plane can expressed as a linear combination of the vectors of the basis
a = a1e1 + a2 e2 = 4e1 + 3e2
e2 a ⎛ 4⎞ ⎛ 1⎞ ⎛ 0⎞
a = ⎜ ⎟ = 4 ⎜ ⎟ + 3⎜ ⎟
⎝ 3⎠ ⎝ 0⎠ ⎝ 1⎠
e1
Wavefunction: if we define a basis (i.e. pz orbitals on C atoms), the benzene molecular orbitals
can be expressed as a linear combination of the vectors of the basis
{
Base (2 p1z ) + (2 pz2 ) + (2 pz3 ) + (2 pz4 ) + (2 pz5 ) + 2( pz6 )}
ψ 1 = 1(2 p1z ) + 1(2 pz2 ) + 1(2 pz3 ) + 1(2 pz4 ) + 1(2 pz5 ) + 1(2 pz6 )
ψ 1 = (1,1,1,1,1,1)
ψ 2 = 1(2 p1z ) + 1(2 pz2 ) + 0(2 pz3 ) − 1(2 pz4 ) − 1(2 pz5 ) + 0(2 pz6 )
ψ 2 = (1,1, 0, −1, −1, 0)
MFQC - Alcamí 4
• Vectors.
• Matrices.
• Determinants.
• N-dimensional complex spaces (Dirac notation).
• Basis exchange.
MFQC - Alcamí 1
, Motivation: Vectors, Matrices and Quantum Chemistry
Real space: a vector in the plane can expressed as a linear combination of the vectors of the basis
a = a1e1 + a2 e2 = 4e1 + 3e2
e2 a ⎛ 4⎞ ⎛ 1⎞ ⎛ 0⎞
a = ⎜ ⎟ = 4 ⎜ ⎟ + 3⎜ ⎟
⎝ 3⎠ ⎝ 0⎠ ⎝ 1⎠
e1
, Motivation: Vectors, Matrices and Quantum Chemistry
Wavefunction. How express it as a vector?
Wavefunction: if we define a basis (i.e. pz orbitals on C
atoms), the benzene molecular orbitals can be expressed
as a linear combination of the vectors of the basis
{ }
Base (2 p1z ) + (2 pz2 ) + (2 pz3 ) + (2 pz4 ) + (2 pz5 ) + 2( pz6 )
ψ 1 = 1(2 p1z ) + 1(2 pz2 ) + 1(2 pz3 ) + 1(2 pz4 ) + 1(2 pz5 ) + 1(2 pz6 )
ψ 2 = 1(2 p1z ) + 1(2 pz2 ) + 0(2 pz3 ) − 1(2 pz4 ) − 1(2 pz5 ) + 0(2 pz6 )
MFQC - Alcamí 3
, Motivation: Vectors, Matrices and Quantum Chemistry
Real space: a vector in the plane can expressed as a linear combination of the vectors of the basis
a = a1e1 + a2 e2 = 4e1 + 3e2
e2 a ⎛ 4⎞ ⎛ 1⎞ ⎛ 0⎞
a = ⎜ ⎟ = 4 ⎜ ⎟ + 3⎜ ⎟
⎝ 3⎠ ⎝ 0⎠ ⎝ 1⎠
e1
Wavefunction: if we define a basis (i.e. pz orbitals on C atoms), the benzene molecular orbitals
can be expressed as a linear combination of the vectors of the basis
{
Base (2 p1z ) + (2 pz2 ) + (2 pz3 ) + (2 pz4 ) + (2 pz5 ) + 2( pz6 )}
ψ 1 = 1(2 p1z ) + 1(2 pz2 ) + 1(2 pz3 ) + 1(2 pz4 ) + 1(2 pz5 ) + 1(2 pz6 )
ψ 1 = (1,1,1,1,1,1)
ψ 2 = 1(2 p1z ) + 1(2 pz2 ) + 0(2 pz3 ) − 1(2 pz4 ) − 1(2 pz5 ) + 0(2 pz6 )
ψ 2 = (1,1, 0, −1, −1, 0)
MFQC - Alcamí 4
