Lecture 6: Coordinates and Basis

The same vector looks different in different coordinate systems.

The Hook

Water Molecule in Two Coordinate Systems

System 1: H₁ = (0.958, 0, 0)

System 2: H₁ = (0.587, 0.758, 0)

Same molecule. Different numbers. The geometry hasn't changed — only our description of it.

Same physical reality. Different numerical description.

How do we translate between them? When does it matter? Why do chemists care?

Recognition

DIRECTION (continued): "It points."

A vector is a direction and magnitude. The components are how we describe it in a particular coordinate system.

Change the coordinate system → change the components → same vector.

This is the distinction between:

The vector (geometric object, coordinate-free)
Its representation (list of numbers, coordinate-dependent)

When to Think About Coordinates and Basis

Situation	What you need
"The components look messy — is there a simpler description?"	Change to a better basis
"I have data in one coordinate system, need it in another"	Change of basis / coordinate transformation
"What do molecular orbital coefficients mean?"	They're components in a basis of atomic orbitals
"Why do quantum calculations depend on basis set?"	Different bases → different representations
"How do I rotate a molecule?"	Rotation matrix (change of coordinates)
"What's the 'natural' coordinate system for this problem?"	Find eigenvectors or symmetry-adapted basis

Basis Vectors

Definition

Definition: A basis for a vector space V is a set of vectors {e₁, e₂, ..., eₙ} such that:

Spanning: Every vector in V can be written as a linear combination of basis vectors
Linear independence: No basis vector can be written as a combination of the others

The number of vectors in a basis is the dimension of V.

The Standard Basis

In ℝ³, the standard basis is:

\mathbf{\hat{i}} = \mathbf{e}_1 = (1, 0, 0)$$ $$\mathbf{\hat{j}} = \mathbf{e}_2 = (0, 1, 0)$$ $$\mathbf{\hat{k}} = \mathbf{e}_3 = (0, 0, 1)

Any vector v = (v₁, v₂, v₃) is written:

\mathbf{v} = v_1 \mathbf{e}_1 + v_2 \mathbf{e}_2 + v_3 \mathbf{e}_3

The components (v₁, v₂, v₃) are the coefficients in this basis.

Explore Different Bases

Basis Explorer

Basis:

Blue: basis vectors | Orange: vector v = (3, 2) in standard

Components in selected basis: (3, 2)

Orthonormal Bases

Definition: A basis is orthonormal if:

All basis vectors are unit vectors: |eᵢ| = 1
All pairs are orthogonal: eᵢ · eⱼ = 0 for i ≠ j

Combined: eᵢ · eⱼ = δᵢⱼ (Kronecker delta)

Why orthonormal bases are nice:

To find the component of v along eᵢ in an orthonormal basis:

v_i = \mathbf{v} \cdot \mathbf{e}_i

Just take the dot product! No matrix inversion needed.

Components in a Basis

Finding Components

Given: Vector v, basis {e₁, e₂, ..., eₙ}

Find: Components (v₁, v₂, ..., vₙ) such that v = Σvᵢeᵢ

Method 1: Orthonormal basis

If the basis is orthonormal:

v_i = \mathbf{v} \cdot \mathbf{e}_i

Method 2: General basis (solve linear system)

Write out the equation v = v₁e₁ + v₂e₂ + ... + vₙeₙ component by component and solve.

This becomes a matrix equation: Ec = v, where E has basis vectors as columns.

Shortcut for orthogonal (but not normalized) basis:

v_i = \frac{\mathbf{v} \cdot \mathbf{e}_i}{\mathbf{e}_i \cdot \mathbf{e}_i}

Change of Basis

Rotation as Change of Basis

Coordinate Rotation

Rotation angle θ: 0°

Original: v = (1.00, 0.50)

Rotated: v' = (1.00, 0.50)

The rotation matrix for angle θ counterclockwise:

R(\theta) = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

Transforms coordinates: v' = R(θ)v

Properties of Rotation Matrices

Orthogonal: R^TR = I (inverse = transpose)
Determinant 1: det(R) = 1 (preserves orientation)
Preserve lengths: |Rv| = |v|
Preserve angles: (Rv)·(Rw) = v·w

Gram-Schmidt Orthogonalization

Given: A set of linearly independent vectors {v₁, v₂, ..., vₙ}

Find: An orthonormal basis {e₁, e₂, ..., eₙ} spanning the same space

Gram-Schmidt Step by Step

Starting vectors: v₁ = (2, 1), v₂ = (1, 2)

The Algorithm

Step 1: Normalize the first vector

\mathbf{e}_1 = \frac{\mathbf{v}_1}{|\mathbf{v}_1|}

Step 2: Subtract projection onto e₁, then normalize

\mathbf{u}_2 = \mathbf{v}_2 - (\mathbf{v}_2 \cdot \mathbf{e}_1)\mathbf{e}_1$$ $$\mathbf{e}_2 = \frac{\mathbf{u}_2}{|\mathbf{u}_2|}

Step 3: For additional vectors, subtract projections onto all previous eᵢ, then normalize.

Chemistry Connection: Molecular Orbitals

A molecular orbital ψ is expressed as a linear combination of atomic orbitals {φ₁, φ₂, ..., φₙ}:

\psi = c_1 \phi_1 + c_2 \phi_2 + \cdots + c_n \phi_n

The atomic orbitals form a basis for the space of molecular orbitals.

The coefficients (c₁, c₂, ..., cₙ) are the components of ψ in this basis.

H₂ Molecular Orbitals

MO type:

Coefficients: (1/√2, 1/√2) in {φₐ, φᵦ} basis

Basis Set Dependence

Why do quantum chemistry results depend on basis set?

A "basis set" (STO-3G, 6-31G*, cc-pVDZ, etc.) is literally a choice of basis functions.

Larger basis → more functions → can represent more detail
Smaller basis → fewer functions → coarser approximation

The true wavefunction exists independent of basis. Our numerical representation depends entirely on which basis we choose.

Chemistry Connection: Normal Modes

Two coupled harmonic oscillators have equations that are coupled in the standard {x₁, x₂} basis.

Transform to normal mode coordinates:

q₁ = (x₁ + x₂)/√2 (symmetric stretch)
q₂ = (x₁ - x₂)/√2 (antisymmetric stretch)

The equations become uncoupled — each normal mode oscillates independently!

The lesson: The right basis diagonalizes the problem.

This is why we seek eigenvectors — they are the natural basis for a given operator.

Chemistry Connection: Crystal Coordinates

In crystallography, positions are often given in fractional coordinates relative to the unit cell vectors a, b, c:

\mathbf{r} = x\mathbf{a} + y\mathbf{b} + z\mathbf{c}

where (x, y, z) are fractions (typically 0 to 1).

Fractional ↔ Cartesian Conversion

x: 0.50

y: 0.50

Fractional: (0.50, 0.50)

Cartesian: (2.00, 2.00) Å

Summary: When to Use What

I want to...	Use...
Find components in orthonormal basis	Dot product: vᵢ = v·eᵢ
Find components in general basis	Solve linear system Ec = v
Rotate a molecule	Rotation matrix R(θ)
Convert crystal coordinates	Basis transformation matrix
Simplify coupled equations	Change to normal mode basis
Make a basis orthonormal	Gram-Schmidt
Understand MO coefficients	They're components in AO basis

Exercises

Exercise 1: Components in a New Basis

Find the components of v = (3, 4) in the orthonormal basis:

e₁ = (1, 1)/√2
e₂ = (1, -1)/√2

Since the basis is orthonormal, use dot products:

v₁ = v·e₁ = (3, 4)·(1/√2, 1/√2) = (3 + 4)/√2 = 7/√2 ≈ 4.95

v₂ = v·e₂ = (3, 4)·(1/√2, -1/√2) = (3 - 4)/√2 = -1/√2 ≈ -0.71

Components in new basis: (7/√2, -1/√2)

Exercise 2: Rotation Matrix

(a) Write the 2D rotation matrix for θ = 45°.

(b) Apply it to the vector (1, 0). What do you get?

(a) R(45°) = [[cos45°, -sin45°], [sin45°, cos45°]] = [[1/√2, -1/√2], [1/√2, 1/√2]]

(b) R(45°)(1, 0) = (1/√2, 1/√2)

Exercise 3: Molecular Orbitals

For H₂, the bonding and antibonding MOs are:

ψ₊ = (φₐ + φᵦ)/√2
ψ₋ = (φₐ - φᵦ)/√2

(a) What are the coefficient vectors for ψ₊ and ψ₋ in the {φₐ, φᵦ} basis?

(b) Show that ψ₊ and ψ₋ are orthogonal (assume φₐ and φᵦ are orthonormal).

(a) ψ₊: (1/√2, 1/√2), ψ₋: (1/√2, -1/√2)

(b) ψ₊·ψ₋ = (1/√2)(1/√2) + (1/√2)(-1/√2) = 1/2 - 1/2 = 0 ✓

Exercise 4: Crystal Coordinates

A cubic unit cell has a = b = c = 4 Å.

(a) Convert fractional coordinates (0.25, 0.5, 0.75) to Cartesian.

(b) An atom at Cartesian (3, 2, 1) Å — what are its fractional coordinates?

(a) (x, y, z)_Cart = (0.25 × 4, 0.5 × 4, 0.75 × 4) = (1, 2, 3) Å

(b) (x, y, z)_frac = (3/4, 2/4, 1/4) = (0.75, 0.5, 0.25)