Symmetry: a blueprint for nature

Josef PaldusDepartment of Applied Mathematics

University of Waterloo

Supported by NSERC

Mesoscopic lecture – MSU – October 2007

• introductory remarks

• some ubiquitous examples of spatial symmetries

• origins of group theory (Galois)

• continuous Lie groups - their origin and birth

and some examples of their role in physics

• unitary groups in molecular electronic structure

calculations

• symmetry breaking

• closing remarks – Lie groups in mathematics

Rough outline:

υ′σ µµετρον(well-proportioned, well-ordered)

SYMMETRY, as wide or as narrow you may define its meaning,is one idea by which man through the ages has tried

to comprehend and create order, beauty, and perfection

(Herman Weyl)

… the importance of digging out the structure from its inessentialtrappings became recognized, and it was noticed that in the

theory of GROUPS,

in pure mathematics, the necessary technique had been developed(Sir Arthur Eddington)

1 Symmetry in mathematics1.1 Mathematical model for symmetry1.2 Non-isometric symmetry1.3 Directional symmetry1.4 Reflection symmetry1.5 Rotational symmetry1.6 Translational symmetry1.7 Glide reflection symmetry1.8 Rotoreflection symmetry1.9 Helical symmetry1.10 Scale symmetry and fractals1.11 Symmetry combinations2 Symmetry in science and technology2.1 Symmetry in physics2.2 Symmetry in physical objects

2.2.1 Classical objects2.2.2 Quantum objects2.2.3 Consequences of quantum symmetry2.3 Symmetry as a unifying principle of geometry

2.4 Symmetry in mathematics2.5 Symmetry in logic2.6 Generalizations of symmetry2.7 Symmetry in biology2.8 Symmetry in chemistry2.9 Symmetry in telecommunications3 Symmetry in history, religion, and culture3.1 Symmetry in religious symbols3.2 Symmetry in Social Interactions3.3 Symmetry in architecture3.4 Symmetry in pottery and metal vessels3.5 Symmetry in quilts3.6 Symmetry in carpets and rugs3.7 Symmetry in music

3.7.1 Musical form3.7.2 Pitch structures3.7.3 Equivalency

3.8 Symmetry in other arts and crafts3.9 Symmetry in aesthetics3.10 Symmetry in games and puzzles3.11 Symmetry in literature3.12 Moral symmetry

Subheadings from Wikipedia

People were fascinated with symmetry from times immemorial

A very incomplete list of AREAS OF APPLICATION

MATH • elementary algebra and geometry (F. Klein, Erlangen program)• algebraic structure theory• Galois algebraic structures • number theory• projective geometry• Fourier analysis (harmonic analysis)• functional analysis • differential geometry (symmetric spaces, etc.)• ODE’s and PDE’s (original Lie theory)

ARTS • music (harmony, counterpoint, campanology, etc.)• ornamental groups (glass windows, Persian carpets, needlework, etc.)• architecture• film animation• Scottish country dancing• braids, plaits, knots, etc.

TECHNOLOGY • coding theory, communication systems (automatic error correcting, etc.)• quality control• electric dynamos• tooling design (wrenches, etc.)• electric networks (homology groups)• weaving industry

SCIENCE • chemistry and physics• crystalography• biology

• atomic, molecular, solid state, nuclear, elementary particles, quantum gravity, etc., etc.

• mechanics, • thermodynamics, • scattering theory, etc., etc.

Bilateral symmetry – Bee pendant from the Heraklion Museum in Crete

SnowCrystals.comSymmetry in nature

Examples of Moorish architectureToledo (synagogue),Granada (Alhambra)

Seville (Alcazar)

17th century “Uschak” runner carpet from Turkey

Translational symmetry

The 7 1The 7 1The 7 1The 7 1----D SpaceD SpaceD SpaceD Space----Group SymmetriesGroup SymmetriesGroup SymmetriesGroup Symmetries

Illustrated by Hungarian Folk NeedleworkIllustrated by Hungarian Folk NeedleworkIllustrated by Hungarian Folk NeedleworkIllustrated by Hungarian Folk Needlework

I. Hargittai and G. Lengyel, J. Chem. Edu. 61, 1033 (1984)

Aristocles alias PLATO (=“broad-shouldered”) 428 or 427 − 347 B.C.

Timaeus (dialogue) ~360 B.C.

Calcidius’ Latin Timaeus translation(a medieval manuscript)

elements (stoicheia)

tetrahedron → fireoctahedron → airicosahedron → watercube → earth[dodecahedron → universe]

Raphael’s Platoin the School of Athens fresco

Consider:

at vertex:

k (2α) = k (1 − ) π < 2 π2j

k ( j – 2 ) < 2 j = 2 ( j – 2 ) + 4

k regular j−gons meeting at each vertex j, k ≥≥≥≥ 3

j−gon angle at a vertex:

j π − 2 πj

= ( 1 − ) π2j

2α =

j k

3 345

4 3

5 3

k − 2 < 4 → k < 6

2(k − 2) < 4 → k < 4

3(k − 2) < 4 → 3k < 10

tetrahedronoctahedronicosahedron

cube

dodecahedron

(k – 2)(j – 2) < 4 j, k ≥≥≥≥ 3

β

αγγγγ

αααα+ + + + ββββ++++ γγγγ = = = = ππππ

αααα = = = = ββββ

Mysterium Cosmographicum(1596, 1921)

Johannes Kepler (1571-1630)

Évariste Galois (1811-1832)

He was the first to use the term “GROUP”(for a group of permutations or a symmetric group)

in his proof of

The impossibility of “Quintic Formula” by radicals(i.e., the fact that the 5th degree equations

are not solvable by radicals)

His work developed into the so-called Galois theory – a major branch of abstract algebra(e.g., it can be used for any polynomial equation to determine whether it has a solution by radicals)

The group theory later developed primarily by Lagrange, Cayley, Cauchy, and Camille Jordan

Extension to continuous groups was started systematically in 1884

Marius Sophus Lie (1842 - 1899)

• His idea was to extend Galois theory foralgebraic equations to differential equations

• He thus created the theory of continuous symmetry by introducing the so-calledtransformation groups, nowadays calledLie groups

• One of his greatest achievements was therealization that these groups may be better understood by “linearizing” them and bystudying the corresponding vector fields(or their infinitesimal generators). Thesehave the structure of what we call todayLie algebras (called by him “infinitesimalgroups”). The term Lie algebra was introduced by Hermann Weyl in 1930.

• He attempted a general classification ofLie algebras, but this turned out to be avery difficult problem.

The history of the development of Lie group theory is very exciting !!!

Felix Christian Klein (1849 - 1925)

• Close friend of Lie – later this friendship suffered due to Lie’s illness• Erlangen program (1872): Classification of geometries by their underlying

symmetry groups – his major contribution• Great pedagogue and encyclopedist – 1890 turned to mathematical physics –

collaborated with Arnold Sommerfeld (gyroscope)

Wilhelm Karl Joseph Killing (1847 - 1923)

• Was first to classify simple Lie algebras• Accused by Lie of stealing his ideas• Also introduced: Killing form

root systemsCartan matricesCartan subalgebras

• Four main infinite series of LGs:A series - SU(r+1) or ArB series - SO(2r+1) or BrC series - Sp(2r) or CrD series - SO(2r) or Dr

plus “exceptional cases”:G2, F4, E6, E7, and E8

There exist exciting relationships between: exceptional Lie groupsnumber fieldspolytopes(especially 3-polytopes, namely Platonic solids)

Parthenon on the Acropolis

l

2l + 1

Basic layout of a classical Greek temple

Temple of Hephaistus (Hephaisteion), Athens

Stoa of Attalos – ancient Athenian Agora

Johan Jacob BALMER (1825-1898)

• teaching mathematics and caligraphyat a secondary school for girls in the city of Basel (Switzerland)

• from 1865, he also lectured at the University of Basel

• His main interest was in numerology:(how many sheep were in a flock, a number of steps of a pyramid, dimensions of local churches, etc.)

• When he complained to his friend, Prof. E. Hagenbach that he had “run out of things to do”, he suggested to look on the spectrum of hydrogen as measured by Ångström

Hδ

3632h

Hγ

2521h

Hβ

1612h

Hα

95 h

• At the time, only four visible lines were known, which he represented in terms of a basic constant “h” (with h = 3645.6 Å):

and recognized the sequences:or

Hδ

98 h

Hγ

2521h

Hβ

43 h

Hα

95 h

for the numerators: 32, 42, 52, 62, … m 2, andFor the denominators: 32−22, 42 −22, 52 −22, 62 −22, … m 2 −22

UV

Hα

95

Hβ

43

Hγ

2521

Hδ

98

Hε

4945

Hζ

1615

Hη

8177

Hθ

2524

Hι

121117

m = 3 4 5 6 7 8 9 10 11

λλλλ = c m2

m2 – 22

C = 3645.6 (mm/107)

1/λλλλ = R Z2 �1/n2 – 1/m2 � n = 2

I gradually arrived at a formula which …

expresses a law by which the wavelengths

of the measured lines can be represented

with striking precission.

Johan Jacob BALMER (1825-1898)

Annalen der Physik und Chemie 25, 80-5 (1885)“Notiz über die Spektrallinien des Wasserstoff”

Symmetry in Physics and Chemistry

SYMMETRY

INVARIANCE

DEGENERACY

GROUPS

DYNAMICAL

NON-INVARIANCE

SPECTRUM-GENERATING

GROUPS or

ALGEBRAS

vs

COMPACT

[SO(n), U(n), Sp(2n), …]NON-COMPACT

[U(1,1), SO(2,1), SO(4,2), …]

Lie Algebras for atomic and molecular structure

• vibron model U(4), U(4)×U(4), …• UGA U(n), U(2), U(2n), …

(a) CI, MR-CI, CAS SCF, …(b) MBPT, GF, …(c) system partitioning(d) spin-dependent Hamiltonians(e) density matrices, etc.

• CAUGA U(n) ⊂ U(2n)(a) VB method(b) CC methods

• H-atom in strong electric and/ormagnetic fields

• H2+ - long-range behaviour

• LOPT• scattering problems• relativistic Kepler problems (with

or without magnetic charges)• electronic “Zitterbewegung”• relativistic oscillators, etc.

Lie groups and algebras

SO(2)

g(ϕ) = eiϕ = cos ϕ + i sin ϕcos ϕ sin ϕ

−sin ϕ cos ϕor

Lie groupgroup

manifold

dg(ϕ)dϕ ϕ = 0

= i

cos ϕ sin ϕ−sin ϕ cos ϕ

exp ϕ =0 1-1 0

or0 1

-1 0and

g(λ1λ2λ3…) = exp ∑i λi εi

d g(λ1λ2λ3…)d λi λ1=λ2=λ3=…=0

= εi

In general:

so(2)

ϕ(1,0)

eiϕ

g - Lie algebra of a Lie group G

g is a vector space (spanned by generators εi ) with an additional operation “••••” (Lie product) s.t. for ∀ a, b, c ∈ g :

(i) closure a • b ∈ g

(ii) bilinearity (xa + yb) • c = xa • c + yb • c

(iii) anti-commutativity a • b = − b • a

(iv) Jacobi identity a • (b • c) + b • (c • a) + c • (a • b) = 0

G = exp (g)

Examples:

(1) R3 (3-D Euclidean space) with the vector (cross) product “××××”a • b a ×××× b

(2) Classical mechanics: L • H {L ,H } (Poisson bracket)

(3) Quantum mechanics (and linear operators):A • B [A, B] ≡ AB − BA (commutator)

Heisenberg’s EOM: [qj, qk] = [pj, pk] = 0, [qj, pk] = i ħ δjk

In general: εi •••• εj ≡ [εi, εj ] = cijk εk structure constants

H = ∑i ai Ci + ∑i ai’ Ci’ + ∑i ai” Ci” + …

G ⊃⊃⊃⊃ G’ ⊃⊃⊃⊃ G” ⊃⊃⊃⊃ … Ci’ ∈ G’, Ci” ∈ G”, etc.

In a strong sense:

Ci ,Ci’, Ci” … (Casimir operators of G, G’, G”, etc.)

DYNAMICAL GROUPS (general)

An entire spectrum of solutions for H is containedin a single irreducible representation (irrep)

In a weak sense:

H = E0 + ∑i ai εεεεi + ∑ij bij εεεεI εεεεj + …

H = E0 + ∑αβ uαβ Eαβ + ∑αβγδ uαβγδ EαβEγδ + ….

εεεεi or Eαβ ∈ G (generators of G, they form a basis of a LA g of G)

(Note: sometimes it is more convenient to work with 1/H )

or

SO(3) SO(2,1) SO(4,2) or SO(2,1)

ROTATIONAL

(rigid rotor)

VIBRATIONAL

(harmonic oscillator)

ELECTRONIC

(hydrogen atom)

∼ j ( j +1) ∼ ( n + ½ ) ∼ 1/n 2

0

1

2

3

4

5

67

012

3

4

5

6

1

2

3456

continuum

(r and pr − a canonically conjugate pair of variables)

Let m = e = ħ = 1, c ≈ 137 and [r, pr] ==== i

and define:

w1 = r

w2 = r pr

w3 = r pr2 + ξ/r

[w1, w2] = i w1

[w3, w2] = −i w3[w1, w3] = 2i w2

T1 = ½(w3 − w1)T2 = w2

T3 = ½(w3 + w1)

[T1, T2] = −i T3

[T2, T3] = i T1

[T3, T1] = i T2

K1 = i T1

K2 = i T2

K3 = T3

so(3) so(2,1)

Casimir’s J 2 = J12 + J2

2 + J32

[J 2, Jk] = 0

C 2 = K12 + K2

2 + K32 = T3

2 − T12 − T2

2

= ξ

[Jj , Jk] = i εεεεjkljkljkljkl Jllll [Kj , Kk] = i εεεεjkljkljkljkl KllllCR’s

Ladder’s J± = J1 ± i J2 T± = T1 ± i T2

[J3 , J±] = ± J± [T3 , T±] = ± T±

We thus have a complete analogy to so(3) let’s play the game

Canonical basis J 2|JM⟩ = J(J+1)|JM⟩

J3|JM⟩ = M|JM⟩

−J ≤ M ≤ J

J 2, J3 C 2, T3spectrum bounded from below

∃|Ψ0⟩ such that T3 |Ψ0⟩ = b0|Ψ0⟩ & T− |Ψ0⟩ = 0

C 2|Ψ0⟩ = T3 (T3 −1)|Ψ0⟩ = b0 (b0 −1)|Ψ0⟩ = ξ|Ψ0⟩

K1 = i T1 = i (w3 − w1)/2K2 = i T2 = i w2

K3 = T3 = i (w3 + w1)/2

and T3 |Ψm⟩ = ( b0 + m )|Ψm⟩ m ≥ 1

Let’s apply this to the H-like atomsH = ½½½½p 2− Z / r

p 2 = pr2 + L 2/r 2 = pr

2 + l (l+1) /r 2

rescale: w1 = λ W1, w2 = W2 , w3 = (1/λ) W3

1 λ = B −−−−½½½½

½(W3 + W1)|Φr⟩ = ½ B −−−−½½½½D |Φr⟩

T3|Φr⟩ = (b0 + m)|Φr⟩

m = 0, 1, 2, …

(r pr2 + l (l+1) /r − 2 r E ) |Ψr⟩ = 2 Z |Ψr⟩

−2 E w1w3

ξ = l (l+1) = b0 (b0 −1) b0 = l+1

thus ½ B −−−−½½½½D = b0 + m = l + 1+ m

and B −1D 2 = 4 (l + 1+ m) 2

= 4 n2

B = ¼D 2 /n2 = −2 E = ¼(2 Z )2 /n2

2rH |Ψr⟩ = E |Ψr⟩

general form: D = 2 Z B = −2 E where(w3 + B w1)|Φr⟩ = D|Φr⟩

λ[(1/λ) W3 + λ B W1]|Φr⟩ = D|Φr⟩

(W3 + λ2B W1)|Φr⟩ = λD|Φr⟩

ergo E = − ½ Z 2 / n2 n ≥ l + 1

w1 = r

w2 = r pr

w3 = r pr2 + ξ/r

ξ = b0 (b0 −1)recall:

T3 |Ψm⟩ = ( b0 + m )|Ψm⟩

SO(3)so(3)

SU(2) ⊂ U(2)su(2)

U(n)

u(n)GRPalgraLie

E11– E22 = 2S3E12 = S+; E21 = S–E11+ E22 = N

Ei,j (i,j=1, …, n)J3; J± = J1 ± i J2Generators

Ei,j† = Ej,i

Structureconstants

[J1, J2] = i J3 (cycl.) [Ei,j, Ek,l] = δjk Ei,l – δil Ek,j

Casimir(s) J 2 = J12 + J2

2 + J32 I (k) = tr(Ek), E = Ei,j

Group chain

SO(3) ⊃ SO(2)abelian

U(n) ⊃ U(n–1) ⊃ ... ⊃ U(1)

abelian

Hermitianproperty

J3† = J3

(J±)† = JŦ

Irreps D (j) (dim D (j) = 2j+1) m1n≥ m 2n

≥ … ≥ mnn

Betweennessconditions

–j ≤ m ≤ j m i,j≤ m i,j-1 ≤ m i+1,j

Basis states j, m ⟩

2j 0j+m

m12 m22m11

=

m1nm 2n

………….. mnn

m1,n-1 ............ mn-1,n-1

..……......m12 m22

m11

≡ [m]

U(n) representation theory (in a nutshell)

• All irreps are finite-dimensional and are labeled by their highest weight

[m1nm 2n

………….. mnn

] m1n≥ m 2n

≥ ….. ≥ mnn

min

∈ Z

n

m1n

m2n

mnn

• Basis states are given by the betweenness conditions m i,j≤ m i,j-1 ≤ m i+1,j

forming the lexically-ordered canonical Gel’fand-Tsetlin (GT) basis

• Explicit expressions may be derived for matrix elements of generators Ei,j

in the canonical GT basis

• Dimension is given by the Weyl formula; schematically [example for U(7) ]

7

77

8 9 10 11

86

5

4

6

2 · 2 · 2 · 3 · 5

6 · 7 · 7 · 7 · 8 · 9 · 10 · 11==== 4 · 7 3 · 9 · 11 ==== 135,828

8 6 5

4

2 1

5 2

2

3

1

1

====dim [5 3 3 1 0] =·

Ecdab = Ec

a Edb – δc

b EdaH = ∑za

b Eba + ∑vab

cd Ecdab

Eστ = ∑eaσ

aτ

a=1

n

U(2n) ⊃⊃⊃⊃ U(n) ⊗ U(2)

Eba = ∑ebσ

aσ

σ=±½eaσ

bτ = X†bτ Xaσ

⊗

↔

a

b= 2S

2S

m1nm 2n

………….. mnn

2 2 … 2 1 1 …..1 0 ….. 0 an bn cnm1,n-1 ............ m

n-1,n-1 2 …… 2 x 1..1 y 0..0 an-1bn-1cn-1..……...... = 2 ………........... ≡ ……m12 m22 v w a2 b2 c2

m11 z a1 b1 c1

[m] ≡

a ≡ an b ≡ bn c ≡ cn

P. Jordan (1935) U(∞)

M. Moshinski (1966)

⟨ [m′] Ej,k [m]⟩ = ∑ ⟨ [m′] Ej,k-1 [m′′]⟩ ⟨ [m′′] Ek-1,k [m]⟩m′′

= ∏ W(bi,ni)i= j

k

Gel’fand-Tsetlin tableau ABC (Paldus) tableau

Matrix elements

a = ½N − S

b = 2S

c = n − a − b = n − (½N + S )

∆ai ,∆ci = 0 or 1

∆bi = 0 or 1 or −1 ai + bi + ci = i

∆ai + ∆bi +∆ci = 1

and

∆ai∆ci

00011011

∆ai∆ci

01001110

∆ai∆bi ∆ci

0 0 10 1 01 –1 11 0 0

so that

0123

di ≡ (∆ai∆ci)2

step numbers

ABC (or Paldus) tableau formalism

2 2 … 2 1 1 …..1 0 ….. 0 an bn cn2 …… 2 x 1..1 y 0..0 an-1bn-1cn-1

2 ………........... ≡ ……. ≡ [abc]v w a2 b2 c2

z a1 b1 c1

[m] ≡

a ≡ an b ≡ bn c ≡ cnRecall

n = a + b + cN = 2a + b

2S = b

where

Example: MBS 4-electron triplets N= n= 4 S= 1

2110 2110 2110 2110 2110 2110 2110 2110 2110 2110 2110 2110 2110 2110211 211 211 210 210 210 210 210 210 210 210 111 110 21 21 11 21 21 20 20 20 11 10 10 11 112 1 1 2 1 2 1 0 1 1 0 1 1

[m]

121 121 121 121 121 121 121 121 121 121 121 121 121 121120 120 120 111 111 111 111 111 111 111 111 030 021 021 110 110 020 110 110 101 101 101 020 011 011 020 020 011 100 010 010 100 010 100 010 001 010 010 001 010 010 010

[abc]

01 01 01 00 00 00 00 00 00 00 00 11 10 00 00 10 01 01 00 00 00 11 10 10 00 01 00 00 10 00 00 10 01 11 10 00 01 00 00 0010 00 00 10 00 10 00 01 00 00 01 00 00

[∆a∆c]

00 0 00 0 00 0 01 1 01 1 01 1 01 1 01 1 01 1 01 1 01 1 10 2 11 301 1 01 1 11 3 00 0 00 0 01 1 01 1 01 1 10 2 11 3 11 3 01 1 00 0 01 1 11 3 01 1 01 1 11 3 00 0 10 2 11 3 01 1 00 0 01 1 01 1 01 111 3 01 1 01 1 11 3 01 1 11 3 01 1 00 0 01 1 01 1 00 0 01 1 01 1

[∆a∆c] di

di ≡ (∆ai∆ci)2

0123

∆ai∆ci ∆ai∆ci ∆ai∆bi ∆ci

00011011

01001110

0 0 10 1 01 –1 11 0 0

recall:

4

3

2

5 4

2

1

12 · 3 · 4 · 5

2 · 4= = 15

2110

211 210 111 110

21 20 11 10

2 1 0

0

01 2

3

Shavit’s graph

U(n)

SNSU(2)

Compact representation via Distict Row Table (DRT)

Distinct Dim

N n S row # FCI

4 4 1 13 153 4 ½ 14 205 6 ½ 32 2106 10 1 80 6 930

10 20 1 355 99 419 40010 30 0 511 4 035 556 161

2 1 1 0 1212 1 0 111

2 1 1101 010

000

00 01 1 101 00 0 010 11 3 200 01 1 1

[m] ABC ∆AC ∆AC di ni

ni = ∆Ni = 2∆ai + ∆bi = ∆ai + ∆ci

1

½½½

0

Si

Yamanouchi-Kotani

couplingscheme

Matrix elements

⟨ [m′] Ej,k [m]⟩ = ∑ ⟨ [m′] Ej,k-1 [m′′]⟩ ⟨ [m′′] Ek-1,k [m]⟩m′′

= ∏ W(bi,ni)i= j

k

S

•

••

•

• ••

•

•

•

•

•

•

•••

½

Sj-1Sj-1

Sj Sj

Sk-1Sk-1

SkSk

Sn Sn

S1S1

S0= 0

• • • • • •• •

• • • ••

•

•

• ••

• • ••

•

•

3j6j

½

½½

Some existing actual exploitations of the unitary group formalism

• Configuration interaction (shell model) calculationstoday many codes available based on this formalism,CI of very large dimension are routine (106 → 109 → ...)

• Density matrix formalismCalculation of reduced density matrices

• Spin-orbit couplingA general formalism for spin-dependent Hamiltonians

• Coupled cluster approachUGA CC enables a fully spin-adapted formalism fortypical open-shell systems

• Phenomenological description of vibronic spectravia Iachello-Levin formalism based on low-dimensionalunitary groups

UGA CCSD

Cluster Ansatz:

Spin-free UGA or CAUGA state (generally multi-determinantal)used as a reference

Orthonormal: but

Excitation operators :

Spin-adapted: linear combinations of generators

Unitarily invariant: (and thus ) are irreducible tensor operatorsadapted to the group chain

Terminology:

Fully SA : (i) must be a pure spin state(ii) gives the same result when applied to any oneof the components with of a given multiplet

Partially SA : is strictly SA, but not

Reference space SA : is SA (e.g., ROHF-type reference), but is

spin-contaminated energy

is “free” of spin contaminants, but computed with

incorrect

Spin non-adapted : is not SA (e.g., DODS UHF) both the

energy and wave function are spin-contaminated

Closed-shell case: Open-shell doublet:

Example: A simple open-shell case

a,b,…

i

r,s,…

BeH X 2∑+ 1.356 1.343 13 2063 2061 2A 2Πr 1.353 1.334 19 2023 2089 -66

BH X 1∑+ 1.244 1.232 12 2357 2367 -10A 1Π 1.239 1.219 20 2191 2251 -60c’ 1∆ 1.194 1.196 -2 2731 2610 121

CH X 2Π 1.131 1.120 11 2826 2859 -33CH+ X 1∑+ 1.128 1.131 -3 2933 [2740] [193]

a 3Π 1.131 1.136 -5 2805 (2814) (-9)A 1Π 1.238 1.234 4 1851 1865 -14b 3∑ - 1.231 1.245 -14 2168 [1939] [229]B 1∆ 1.218 1.233 -15 2169 2076 93

NH X 3∑ - 1.048 1.036 12 3234 3282 -48NH+ X 2Π 1.072 1.070 2 3104 [2922] [182]OH X 2Π 0.983 0.970 13 3658 3738 -80

A 2∑+ 1.019 1.012 7 3130 3179 -49OH+ X 3∑ - 1.041 1.029 12 3087 3113 -26FH+ X 2Π 1.031 1.001 30 2991 3090 -99

A 2∑+ 1.267 1.224 43 1431 1496 -65

System State UGA-CCSD Exp. Diff. UGA-CCSD Exp. Diff.

Re (in Ǻ; Diff. in 10-3 Ǻ) ωe (in cm-1)

UGA-CCSD - 1st row diatomic hydrides – 6-31G* basis

BeH X 2∑+ 1.356 1.343 13 2063 2061 2A 2Πr 1.353 1.334 19 2023 2089 -66

BH X 1∑+ 1.244 1.232 12 2357 2367 -10A 1Π 1.239 1.219 20 2191 2251 -60c’ 1∆ 1.194 1.196 -2 2731 2610 121

CH X 2Π 1.131 1.120 11 2826 2859 -33CH+ X 1∑+ 1.128 1.131 -3 2933 [2740] [193]

a 3Π 1.131 1.136 -5 2805 (2814) (-9)A 1Π 1.238 1.234 4 1851 1865 -14b 3∑ - 1.231 1.245 -14 2168 [1939] [229]B 1∆ 1.218 1.233 -15 2169 2076 93

NH X 3∑ - 1.048 1.036 12 3234 3282 -48NH+ X 2Π 1.072 1.070 2 3104 [2922] [182]OH X 2Π 0.983 0.970 13 3658 3738 -80

A 2∑+ 1.019 1.012 7 3130 3179 -49OH+ X 3∑ - 1.041 1.029 12 3087 3113 -26FH+ X 2Π 1.031 1.001 30 2991 3090 -99

A 2∑+ 1.267 1.224 43 1431 1496 -65

System State UGA-CCSD Exp. Diff. UGA-CCSD Exp. Diff.

Re (in Ǻ; Diff. in 10-3 Ǻ) ωe (in cm-1)

UGA-CCSD - 1st row diatomic hydrides – 6-31G* basis

UGA-CCSD results for some1st row diatomics

6-31G* basis

Mean absolute deviations:0.012 Å and 73 cm-1

Maximal deviations:0.028 Å and 203 cm-1

Symmetry breaking

(spontaneous or otherwise)

Symmetry-adapted (SA)solution

Symmetry breaking

(spontaneous or otherwise)

Broken-symmetry (BS)solution

Limburg,

Gemany

Speyer,

Germany

Regensburg,

Germany

Chartres,France

1 Kings 3:16−27

C

H

C

H

C

H

C

H

C

H

C

H

C

H

C

H

C

H

C

H

C

H

H H H H H H H H H

N = 14 N = 6

all-trans polyacetylene(∞ linear polyene)

Using Born – von Kármán cyclic boundary conditions cyclic polyene model

N = 2n = 4ν + 2 non-degenerate ground stateChoosing

benzene ring(π−electron model)

Planar π-electron model systems with conjugated double-bonds

PPP (Pariser-Parr-Pople) Hamiltonian

N = 14

p-anthracene

H = β∑ X†j Xk + ∑ γjk X†

j X†k Xk Xj

j,k j,k

n.n.

Coulomb integralresonance (hopping) integral

∼ (coupling constant)-1

cyclic polyenes (rings)p-polyacenes

linear polyacenes

N = 4ν + 2

Theorems: • Any Kekulé or Dewar solution represents an exact RHF solutions in the fullycorrelated limit of β = 0.

• The total π-electron energy of any Dewar solution is higher than the energyof any Kekulé solution.

• In the fully correlated limit β = 0, all Kekulé solutions are degenerate and their energy represents the absolute minimum of the energy in thevariational space that is spanned by all single determinantal wave functionswith doubly occupied orbitals.

Let the symmetry of the π-electronic Hamiltonian of a system with conjugated double bonds be characterized by a point group G.

If there exists a Kekulé structure having the same point group symmetry G,then the SA RHF solution of a given system is always stable.

However, if all the Kekulé structures have a lower symmetry H, where H designatesa proper subgroup of G, then the SA RHF solution may become singlet unstablein some region of the coupling constant, namely for 0 ≤ |β| < βcrit .

Corollary:

Some basic theorems

PPP model

π−electron energy Eπ as a function

of the variational parameter τ for β= −0.1 eV for undistorted (γ = 1)

and several distorted (γ = d0/d1) cases.

N= 6 N= 26

π and σ−electron RHF and UHFenergy as a function of

dimensionless distortionparameter ∆e/Re

PECs for distorted polyenes CNHN,N=4ν+2, as a function of ∆e/Re or γ.

ν =

Dependence of ∆e (in Å)and of the stabilizationenergy per site ∆ε/N on thesize of the polyene N

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5

6

7

8

9

10

SA and BS RHF solutions for p-naphthalene (or 10-annulene)

1

2

3

45

6

7

8 9

10

11 12

1314

(a) UHF solution for N3(b) ROHF solution for N3(c) BNB case: SA - symmetry adapted ROHF solution

BS - broken symmetry ROHF solution

(a) (b) (c)

SA

Typical HF PESs for linear ABA species

R – symmetric stretching mode∆ − asymmetric stretching mode

BS

E

(cm-1)

∆ (Å)

CCSD

CCSD(T)

Harmonic PEC

v = 0

v = 1

v = 2

v = 3

v = 4

PECs for the ground state of N3along the asymmetric stretching coordinate ∆

Harmonic PECcorresponds toexperimental asymmetricstretching

frequency of1644.68 cm-1

E

(cm-1)

∆ (Å)

PECs for the ground state of N3along the asymmetric stretching coordinate ∆

v = 0

v = 1

v = 2

v = 3

v = 4

UCCSD

UCCSD(T)

FCI

ROHF UHFBNB

PECs along the antisymmetric stretching coordinate ∆as obtained with 2R RMR CCSD(T)

E

(cm−1)

∆ (Å)

cc-pVQZ

cc-pVDZR1 = R1e + ∆

R2 = R2e − ∆

R1e

cc-pVDZ

cc-pVQZ

1.303

1.285

R2e

1.386

1.368

Calculated and experimentally determined vibrational energies ∆E(ν)relative to the zero point energy E(0) [given in parentheses] for thefirst seven levels of the antisymmetric stretching mode ν3 of BNB

012345

6

0 (332)8852085343148976524

8354

0 (497)7522016331747896416

8246

882.31998.3(?)d

3330

58826123(?)d

85520523291

5888

0 (353)8362011327146266042

7514

0 (370)8712089339348026291

7874

v A B IR(mtx) PES cc-pVDZ cc-pVQZ

(±40 cm−1)

BD(T)/cc-pVDZa experimentb RMR CCSD(T)c

a Based on a BD(T)/cc-pVDZ potential whose singular behavior at Q3=0 was replaced by a constant potential interconnecting two broken-symmetry minima (case A) or viaa polynomial interpolation resulting in a central hump (case B) – see Asmis et al. paper.

b Matrix IR spectrum (as interpreted by Asmis et al.) and anion photoelectron spectrum. c Computed with the LEVEL codes using RMR CCSD(T) potentials (see figure).d Question mark indicates an uncertain assignment.

Timaeus ~360 B.C.elements (stoicheia)

tetrahedron → fire (simplicity and sharp corners)octahedron → air (spins nicely in the air if you hold it

between finger and a thumb)icosahedron → water (“round” – densest and least

penetrating of three)cube → earth (most immovable and plastic

of the four)[dodecahedron → universe]

Note: • C60 is a truncated icosahedron(one of 13 Archimedean solids)

• many viruses are icosahedrons• molecules known of all 5 structures

• icosa- and dodeca-hedrons cannot fill the space,but distorted ones can (trivalent rare earth ions –LaMg double nitrates, etc.)

PLATONIC SOLIDS … bodies that excel in beauty …

A mathematician’s perspective

Let’s go back to:

real #s R 1 1complex #s C 2 1, i (i 2= −1) algebraically closedquaternions H 4 1, i, j, k (i 2= j 2= k2= ijk= −1) noncommutativeoctonions O 8 1, e1, …, e 7 (Fano plane) nonassociativesedenions ? 16 (Cayley-Dickson process) not a division algebra !!!

But: can form tensor products with octonionsR ⊗⊗⊗⊗ O octonionsC ⊗⊗⊗⊗ O bioctonionsH ⊗⊗⊗⊗ O quateroctonionsO ⊗⊗⊗⊗ O octooctonions

faces vertices edges grp order

tetrahedron 4 4 6 S4 24

cube 6 8 12octahedron 8 6 12

dodecahedron 12 20 30icosahedron 20 12 30

E6

E7

E8

McKay correspondence(a remarkable coincidence

between the symmetry groups ofPlatonic solids [i.e., subgroups

of SO(3)] and the root systems of simply-laced exceptional complex

simple Lie groups (algebras)

Number systems (normed division algebras)Relationship with:

Duality property: Recall Euler’s theorem: V + F = E + 2

S4 × Ι 48

A5 × Ι 120

Now:

G2 14 group of symmetries of O

F4 52 OP2

E6 78 (C ⊗⊗⊗⊗ O) P2

E7 133 (H ⊗⊗⊗⊗ O) P2

E8 248 (O ⊗⊗⊗⊗ O) P2

group of isometries of

grp dim relationship Dynkin (Coxeter) diagram

A group symmetry for heterotic string theory(a mixture of bosonic string and superstring

[≡ supersymmetric string] theories)

Atlas project (March 2007):(an intensive 4 year collaboration of

18 mathematicians from U.S. and Europe)

• structure and all representations for E8 • results required 60 GB of memory –

(human genome requires < 1 GB)

The mathematician plays a game in which he himself invents the rules

while the physicist plays a game in which the rules are provided by nature,

but as the time goes on it becomes increasingly evident that the rules

which the mathematician finds interesting are the same as those whichnature has chosen. (Paul Adrien Maurice Dirac)

God ever geometrizes(Plato)

… the enormous usefulness of mathematics in the natural sciences is

something bordering on the mysterious and there is no rational explanation for it.

(Eugene Wigner)

Thank you very much !!!