Supplementary information for

Supplementary information for

“Aiming at an Accurate Prediction of Vibrational and Electronic Spectra for Medium-to-Large Molecules: An Overview”

Julien Bloino ^∗ , Alberto Baiardi ^† , Malgorzata Biczysko ^‡

April 20, 2016

1 DARLING-DENNISON RESONANCES

1.1 1-1 Terms: h v + 1 _i | H ⁰ | v + 1 _j i

h v + 1 i | H ⁰ | v + 1 j i = p(v _i + 1)(v _j + 1) 2





 3

2 (v i + 1)K ii;ij + 3

2 (v j + 1)K ij;jj +

N

X

k=1k6=i,j

 v k + 1

2

 K ik;jk





 (1)

with

K ii;ij = 1 6 k iiij

− 1 24

N

X

l=1

k ijl k iil

 1

2ω i + ω l

+ 1

ω l − 2ω i

+ 4 ω l

− 2

ω i − ω j − ω l

− 2

ω j − ω i − ω l

+ 1

ω i + ω j + ω l

+ 1

ω l − ω i − ω j



(2)

K ij;jj = 1 6 k ijjj

− 1 24

N

X

l=1

k _ijl k _jjl

 1

2ω j + ω l

+ 1

ω l − 2ω j

+ 4 ω l

− 2

ω j − ω i − ω l

− 2

ω i − ω j − ω l

+ 1

ω i + ω j + ω l

+ 1

ω l − ω i − ω j



(3)

K ik;jk = 1

2 k _ijkk + 2 X

τ

B _τ

^eq

ζ _ik,τ ζ _jk,τ (ω _k ² + ω _i ω _j ) ω k

√ ω i ω j

− 1 8

N

X

l=1

k _ikl k _jkl

 1

ω i + ω k + ω l

+ 1

ω l − ω i − ω k

+ 1

ω j + ω k + ω l

+ 1

ω l − ω j − ω k

− 1

ω i − ω k − ω l

− 1

ω k − ω i − ω l

− 1

ω j − ω k − ω l

− 1

ω k − ω j − ω l



− 1 8

N

X

l=1

k _ijl k _kkl



− 1

ω j − ω i − ω l

− 1

ω i − ω j − ω l

+ 2 ω l



(4)

∗

Consiglio Nazionale delle Ricerche, Istituto di Chimica dei Composti OrganoMetallici (ICCOM-CNR), UOS di Pisa, Area della Ricerca CNR, Via G. Moruzzi 1, 56124 Pisa, Italy

†

Scuola Normale Superiore of Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy

‡

International Centre for Quantum and Molecular Structures, College of Sciences, Shanghai University, 99 Shangda Road, Shanghai, 200444 China

1

1.2 2-2 Terms: h v + 1 _i + 1 _j | H ⁰ | v + 1 _k + 1 _l i

1.2.1 2-2 Terms

h v + 2 i | H ⁰ | v + 2 k i = p(v i + 1)(v i + 2)(v k + 1)(v k + 2)

4 K ii;kk (5)

with

K _ii;kk = k _iikk 4 − X

τ

B _τ

^eq

ζ _ik,τ ² (ω _i + ω _k ) ² ω i ω k

− 1 16

N

X

m=1

k _iim k _kkm

 1

2ω i + ω m

+ 1

ω m − 2ω i

+ 1

2ω k + ω m

+ 1

ω m − 2ω k



− 1 4

N

X

m=1

k ijm 2



− 1

ω k − ω i − ω m

− 1

ω i − ω k − ω m



(6)

1.2.2 2-11 Terms

h v + 2 i | H ⁰ | v + 1 k + 1 l i = p(v _i + 1)(v _i + 2)(v _k + 1)(v _l + 1)

4 K ii;kl (7)

with

K ii;kl = k iikl

2 − 2 X

τ

B _τ

^eq

ζ ik,τ ζ il,τ (ω i + ω k )(ω i + ω l ) ω i

√ ω k ω l

+ 1 4

N

X

m=1

k ikm k ilm

 1

ω k − ω i − ω m

+ 1

ω i − ω k − ω m

+ 1

ω l − ω i − ω m

+ 1

ω i − ω l − ω m



− 1 8

N

X

m=1

k iim k klm

 1

2ω i + ω m

+ 1

ω m − 2ω i

+ 1

ω k + ω l + ω m

+ 1

ω m − ω k − ω l



(8)

1.2.3 11-11 Terms

h v + 1 i + 1 j | H ⁰ | v + 1 k + 1 l i = p(v i + 1)(v _j + 1)(v _k + 1)(v _l + 1)

4 K ij;kl (9)

with

K ij;kl =k ijkl + 2 X

τ

B _τ

^eq

ζ ij,τ ζ kl,τ (ω i − ω j )(ω l − ω k ) − ζ ik,τ ζ jl,τ (ω i + ω k )(ω l + ω j ) − ζ il,τ ζ jk,τ (ω i + ω l )(ω k + ω j )

√ ω _i ω _j ω _k ω _l

− 1 4

N

X

m=1

k ijm k klm

 1

ω _k + ω _l + ω _m + 1

ω _m − ω _k − ω _l − 1

ω _i + ω _j + ω _m − 1 ω _m − ω _i − ω _j



+ 1 4

N

X

m=1

k _ikm k _jlm

 1

ω _k − ω i − ω m

+ 1

ω _i − ω k − ω m

+ 1

ω _j − ω l − ω m

+ 1

ω _l − ω j − ω m



+ 1 4

N

X

m=1

k _ilm k _jkm

 1

ω l − ω i − ω m

+ 1

ω i − ω l − ω m

+ 1

ω k − ω j − ω m

+ 1

ω j − ω k − ω m



(10)

Note that if the quartic force constants are obtained by numerical differentiation of analytic second derivatives of the energy, then k ijkl is unknown.

2

2 TRANSITION MOMENTS

2.1 1-quantum Transition

h P i 0,1

_i

= 1

√ 2 P i + 1 4 √ 2

N

X

j=1

P ijj − 1 8 √

2

N

X

j,k=1

k ijkk P j

 1

ω _i + ω _j − 1 − δ ij

ω _i − ω _j



− 1 8 √ 2

N

X

j=1 N

X

k=1

 k ijk P jk

 1

ω i + ω j + ω k

− 1

ω i − ω j − ω k



+ 2k jkk

ω j

P ij



+ 1 2 √ 2

N

X

j,k=1

 X

τ =x,y,z

B

^eq

_τ ζ _ik,τ ζ _jk,τ

 P _j

( √ ω i ω j

ω k

 1

ω i + ω j

+ 1 − δ _ij ω i − ω j



− ω _k

√ ω i ω j

 1

ω i + ω j

− 1 − δ _ij ω i − ω j

 )

+ 1 16 √

2

N

X

j,k,l=1

k ikl k jkl P j

( 4ω j (ω k + ω l )(1 − δ ij )(1 − δ ik )(1 − δ il ) (ω _j ² − ω i 2 ) [(ω _k + ω _l ) ² − ω i 2 ] + (ω _k + ω _l ) (ω k + ω _l ) ² − 3ω _i ²  δ ij (1 + δ _ik )(1 − δ _il )

ω i [(ω k + ω l ) ² − ω i 2 ] ² + 4ω j (3ω k + 4ω i )(1 − δ ij )(1 − δ ik )δ il

ω k (ω j 2 − ω i 2 )(ω k + 2ω i ) )

+ k ijk k llk P j

( δ ij

ω i ω k



1 + 2δ ik δ il

9



+ 4ω j (1 − δ ij )(1 − δ ik )(1 − δ il ) ω k (ω j 2 − ω i 2 ) + 4ω _j δ _ik (1 − δ _ij )

ω i (ω j 2 − ω i 2 )

 1 + 2δ _ij

3

 )

(11)

2.2 2-quanta Transition

h P i _0,(1+δ

_ij

₎

_i

_(1−δ

_ij

₎

_j

= 1 2(1 + δ _ij )

"

P ij + 1 2

N

X

k=1

k ijk P k

 1

ω _i + ω _j − ω _k − 1 ω _i + ω _j + ω _k

 #

(12)

3 FRANCK-CONDON INTEGRALS

3.1 0-0 transition integral

h ¯ 0 | ¯ ¯ 0 i =2 ^N/2 det  ¯ Γ ¯ Γ ¯  1/4

[det(J ) det(Y )] ^1/2

× exp



− 1

2 K ^T ΓK + ¯ 1

2 K ^T ΓJ T J ¯ ^T ΓK ¯ ¯

 (13)

with

Y = (J ^T ΓJ + ¯ ¯ Γ) ¯ ⁻¹

3.2 Recursion relations

h ¯ v | ¯ ¯ v i = 1

√ 2¯ ¯v i

h

D i h ¯ v | ¯ v − 1 ¯ i i +

N

X

j=1

q

2(¯ ¯v j − δ ij )C ij h ¯ v | ¯ v − 1 ¯ i − 1 j i +

N

X

j=1

p¯ v j E ij h ¯ v − 1 j | ¯ v − 1 ¯ i i i

(14)

h ¯ v | ¯ ¯ v i = 1

√ 2¯ v i

h

B i h ¯ v − 1 i | ¯ v i + ¯

N

X

j=1

q

2(¯ v j − δ ij )A ij h ¯ v − 1 i − 1 j | ¯ ¯ v i +

N

X

j=1

p¯ ¯v j E ji h ¯ v − 1 i | ¯ v − 1 ¯ j i i

(15)

3

with

A = 2 ¯ Γ ^1/2 J Y J ^T Γ ¯ ^1/2 − I B = −2 ¯ Γ ^1/2 J Y J ^T Γ − I ¯
K C = 2 ¯ Γ ¯ ^1/2 Y ¯ Γ ¯ ^1/2 − I

D = −2 ¯ Γ ¯ ^1/2 Y J ^T ΓK ¯ E = 4 ¯ Γ ¯ ^1/2 Y J ^T Γ ¯ ^1/2

4