2E02
非調和ポテンシャルに対する振動座標の最適化	  	
  （イリノイ大学化学科）	
  八木清，平田聡	
  
	
  
	
  

Optimal	
  vibrational	
  coordinates	
  for	
  anharmonic	
  potential	
  (University	
  of	
  Illinois	
  at	
  Urbana-­‐Champaign)	
  Kiyoshi	  Yagi	  and	  So	  Hirata	
  

	
  
[序]	
  非調和振動状態計算は，厳密解を与える Full	
   CI 計算を除き，座標系の取り方に依存する．現在，座標系には基準座標が最もよく用いられるが，基準座標が必ずしも厳密解への速やかな収束を与えるとは限らない．基準座標は 2 次のモード間カップリングを消すが，非調和カップリングを小さくする保証はないからである．例
えば，非局在化した基準座標より局在化座標の方が X-­‐H 伸縮振動モード（X	
  =	
  O,	
  N,	
  C）
の振動状態を良く記述できることが昔から知られている[1,2]．しかし，任意の系に対し最適な振動座標を求める方法はこれまでになかった．	
  	
  本研究において，我々は vibrational	
   SCF	
   (VSCF)エネルギーを座標変換に対して最小化することで振動座標を求める新しい方法を開発した．この方法では，調和ポテ
ンシャルにより相互作用する振動子に対して基準座標が得られ，一方，相互作用のない非調和振動子（2 原子分子）の集合に対して各振動子に局在化した振動座標が得られる．すなわち，本方法は 2 つの極限において厳密解が容易に得られる座標系を与える．従って，一般の分子系でも良い座標系が得られることが期待できる．得
られた座標を用いて vibrational	
   CI	
   (VCI)計算を実施し，従来の基準座標に基づく計算よりも速やかに厳密解へと収束することを示す．	
  
	
  
[方法]	
  振動回転相互作用を無視すると，振動ハミルトニアンは基準座標 Q により以下の式で与えられる，	
  
f
1
"2
(1)	
  
Ĥ = ! ! 2 +V (Q) ,	  
2 i=1 "Qi
但し f は自由度の数，V(Q)は Born-­‐Oppenheimer ポテンシャルエネルギー曲面であり，平衡点の周りでのテイラー展開で表す．基準座標を変換し，新しい座標系を定義す
る，	
  
f

Q! i = !U siQs ,	
  

(2)	
  

s=1

! に対し，全波動関数を 1 座標関数の直積の形
但し	
   U は直交行列である．得られた Q
で表し，	
  	
  
f

! = " ! (i) (Q! ) ,	  ! nVSCF (Q)
ni
i

(3)	
  

i=1

変換行列と 1 座標関数を同時に最適化する．1 座標関数と VSCF エネルギーは，	
  ff%" 1 "2( j)( j)$!
+ ! !n j V ! !n j '!n(i)i = ! n(i)i !n(i)i ,	
  
(4)	
  
2
!
'&
$# 2 "Qi
j!ij!i

EnVSCF = ! nVSCF Ĥ ! nVSCF ,	
  

(5)	
  

: Q2 : &'(&)*$+,-'$

HFHF
stretchingmodes
aredelocalized
shown
in Figure3.
norma
to eachmolecule.
λmolecule.
derivedAppendix
B,The
is calto
each HF
λin
AppendixB,coordinates
overinthe
twoHFmo
0 , are0 , derived

-'$
4
7over
4
coordinatesaretwomoleculesre: Q̃2 : &'(/)&$+,
culatedto be
9.0×10for9.0×10
Q̃Q̃2 ,Q̃theand
4.6×10forQ̃3symmculatedto7delocalized
be
for
and
Q̃HF
, system:
and4.6×10f
flecting
theC1 andsymmetry
of
the

1
2
2v
: Q2 : &'(&)*$+,-'$: Q̃2 : &'(/)&$+,-'$
flecting
thesymmetryof thesystem:andindicating
advantageof localsymmetric
HF
and
Q̃C
indicating
thecoordinates
advantage
localedge
HF
asymmetricstretchingof coorthe
4 ,2valso the: Q4 : &011)*$+,-'$ : Q4 : &011)*$+,-'$ : Q̃4 : &'.*)&$+,-'$: Q̃4 : &'.*)&$+,-'$and Q̃4 , also

asymmetric stretching coordinates of the edge molecule

dinates. Therefore,
theoptimizationof themolecules
VSCF
energydinates.
Therefore,
theoptimizationof theVSCF
e(Q1 and
Q
the
inner
(Q
2 ) and
4 and
-'$
-'$
(Q1 andQ2 )otherand theinnermoleculesQ). areOn
4 andfor3weakly&'.*)&$+,: Q3Normal: -'$&'./)&$+,
automaticallygenerates
local
coordinatesfor(Qweaklyand: Q̃:-'$optimized
automaticallygenerates
local
coordinates
the
hand,
the
optimized
coordinates
3
FIG. 3. (ColorFIG.
online)(a)and(b)
optimized
coor3.
(Coloronline)(a)Normaland(b)coor: Q3 : &'./)&$+,: Q̃3 : &'.*)&$+,
the other
the molecule.
optimized
coordinates
areAppendix
localized
non-interacting
anharmonic
oscillators.
non-interacting
anharmonic
tohand,each HF
λderived in
dinates of weaklyinteracting
HFinteractingaligned inHF
onealigned
dimension.
dinatesof weakly
in one dimension.
0 ,oscillators.
to
each
HF
molecule.
λ,derivedinAppendixB,
is cal70
The zero-point
energy(ZPE)
and(ZPE)thefundamental
freThezero-point
energyEach HF is separated
byis3separatedÅ.
culatedto be
for Q̃
andtheQ̃2fundament, andEach HF
by 3 Å.
1and
7 9.0×10
4 4.6×culatedto be by9.0×10forQ̃compared4.6×10for thoQ̃
1 and Q̃
2 , and-'$-'$quencies
obtained
VSCF
are
with
those
by
quencies
obtained
by
VSCF
are
compared
with
and
Q̃
,
also
indicatingtheadvantageof
local3&011)*$+,&'.*)&$+,
4: Q̃4 :: Q4 :and Q̃4 , also indicating
the advantage of local HF coor
: Q̃4 : &'.*)&$+,-'$: Q4 : &011)*$+,-'$

により与えられる．VSCF エネルギーは座標に依存するため，これを座標変換に対しVCI[4]-(8) in
Table
I. in
VCI[4]-(8)basedon thetwoon
coorVCI[4]-(8)Table I. VCI[4]-(8)
based
the
dinates.
Therefore,
the optimizationofthe twoVS
dinates.
Therefore,
optimizationof theVSCF energyて最小化することで座標を最適化できる．変換行列に対するエネルギーの最小化はdinates
is foundto isbefoundinthe
agreement
withinaccuracyofdinatestobe in agreement
withinaccuraTABLE I. The TABLE
zero-point
energy(ZPE)(a)andNormalthe(ZPE)
fundamental
I.(ColorThezero-point
energyand(b)
theoptimized
fundamental
automaticallygenerates
local coordinates
for w
FIG.
3.
online)andcoorautomaticallygenerates
local
coordinatesforweakly−1−1 Normal and (b) optimized coor−1 selection of coordinates is not an is- andFIG.(in3.frequencies(Coloronline)frequenciescm) ofofweaklyinteracting
HF
calculated
bycalculated
(in
cm(a)
) of weaklyinteracting
by−1 . Thus,cm
0.1non-interacting
cmthe. Thus, theanharmonic
selection of oscillators.
coordinates is notdinates
weaklyinteracting
HFalignedHF
in
one 0.1
dimension.
non-interacting
anharmonic oscillators.
Jacobi	   sweep
アルゴリズムにより数値的に実行できる．アルゴリズムの詳細は当日
dinatesofweaklyinteracting
HFalignedinone
dimension.
VSCF based on
normalandcoordinates,
the
VSCF
based
onoptimized
normalbyand
optimized andcoordinates,
and forthethe high
sue
level
calculation.
However,thelowthelevel
sue forthezero-point
high level energy
calculation.
However,
the
low
The
(ZPE)
and
fundamEach
HF
is separated
3 Å.
The
zero-point
energy
(ZPE)
and
the
fundamental fre
Each
HF
is separated
3 Å.
deviations
from
VCI[4]-(8)in byparentheses.
deviations
from
VCI[4]-(8)in parentheses.
method (=VSCF)
does
dependonthecoordinates.
The witmethod(=VSCF)
doesoncoordinates
quencies
obtained
bydepend
VSCF
arethecompared
発表する．	
  quencies obtained by VSCF are compared with those by
VSCFb energies
based
on normal
coordinates
are found
VSCF
energies
based
normal coordinates
are
VCI[4]-(8)Table onI.
VCI[4]-(8)oncoorthe
VSCFa
VCI[4]-(8)b VCI[4]-(8)VSCFa
VCI[4]-(8)in Table in
I. VCI[4]-(8)based on based
the two
得られた座標を用いて，振動状態計算を実行する．VSCF
はあくまで平均場レベル
with
large
deviations
witha meanabsolute
withlarge isdeviations
witha agreement
mean deviation
absolute
dev
dinates
found
to
be
in
withinTABLE
I.NormalThezero-point
energy(ZPE)
and the fundamental
Normalcoord.
Optimized
coord.
coord.
Optimized
coord.
dinates is foundagreement within accuracy ao
TABLE I. The zero-point energy(ZPE) and the fundamental−1to be in −1−1(MAD) byof 117.4cm.. Thus,
In strike
those
based
(MAD)
cm
In strike ofcontrast,
thoseis(in) of(0.4)
weaklyHF calculated−1
cmof−1117.4
the. contrast,
selection
coordinates
ZPE frequencies
8419.5frequencies(80.9)
8338.98338.5
ZPE
8419.5(80.9)
8338.9interacting(0.4)
8338.5であり，分光学的精度を得るには相関理論へ進む必要がある．本研究では
VCI計算(in cm) ofcmweaklyinteracting
HF
calculatedby
0.1 cm−10.1. Thus,theselectionofcoordinates
is not an is
VSCF
on andnormalandoptimized
coordinates,
the oncoordinates
on andoptimized
are inlevel
excellentwith thoptimized
coordinates
are inagreement
excellentagreemen
4125.74075.6(1.2)
4074.4
ν1 VSCF4125.7(51.3)
4075.6(1.2)
4074.4
ν1(51.3)
sue
for
the
high
calculation.
However,based
onbasednormaloptimized
coordinates,
and thesue for the−1high level calculation.
However, the low leve
−1
from
VCI[4]-(8)in parentheses.
を実行し，基準座標と最適化座標の比較を行った．以下では最適化座標に基づく
VCI
MAD of 1.5MADcm
This
clearlydemonstratetheν2 deviations
4136.2deviations
4075.6(1.0)
4074.6of. 1.5
cmresult. Thisclearlydemonstraνfrom4136.2(61.6)
4075.6(1.0)
4074.62(61.6)
method(=VSCF)
doesresult
dependon thecoordin
VCI[4]-(8)in parentheses.
method (=VSCF)
does dependonthe coordinates.
Theν34278.4 ν(193.5)
4087.04084.94278.4(193.5) (2.1)
4087.0(2.1)
4084.9impactofcoordinate
optimization
which
dramaticallyenimpactofcoordinate
optimization
which
dramatica
3ab
VSCF energies
based
oncoordinatesnormalcoordinates
a VSCFb
VSCF
energies
basedonnormalare
found
VCI[4]-(8)を oc-­‐VCI と表記する．	
  VSCF
VCI[4]-(8)ν44289.3 ν(199.8)
4087.0(2.5)
4089.54289.3(199.8)4087.0(2.5)
4089.54
hances the accuracy
ofaccuracy
VSCF.
hances
the
with
large
deviations
with absolutea mean absolute
Normalcoord.
coord.
with large
deviations
withof aVSCF.
mean
deviationcNormalcoord. 117.4
Optimized
coord. 1.5
MADc
117.4
1.5 Optimized
MAD
−1
	
  
−1
(MAD)
of
117.4
cmstrike. In
strike contrast,
th
(a)
ZPE
8419.5(80.9)
8338.9(0.4)
8338.5
(MAD)
of
117.4
cm
.
In
contrast,
those based
ZPE
8419.5 (80.9) 8338.9(0.4)
8338.5on optimized
coordinates
are in excellent
a Optimized for atheν1ground(51.3)
4075.6(1.2)4074.4
4074.4on optimized
coordinates
are in excellent
agreementagreewithstate.
fundamentals
theOptimized
for量体の分子内振動モー
theThegroundstate.
The
fundamentals
are the[結果]	
  開発した方法を水2(51.3)4125.7 4125.7
4075.6(1.2)are
ν1
B.
Water
dimer
Water
1.5
cm−1
. Thisresultdemonstrate
clearly demon
ν2the modals.
4136.2(61.6)(1.0)4074.6 B.4074.6
virtual energies
ofvirtualenergies
modals.4075.6
MADdimerof MAD
1.5
cmof−1. Thisresultclearlytheν24136.2(61.6)of the4075.6
(1.0)
b The
b The
ドへ適用し
たresults
．νポ
テ
ン
シ
ャ
ル
エ
ネ
ル
ギ
ー
曲
面
は
ν
4278.4(193.5)4087.0(2.1)
4084.9basedonnormalandoptimizedareinagreement
results
basedonnormalandoptimizedareinagreement
impactofcoordinate
optimization
which drama
34278.4(193.5)4087.0(2.1)
4084.9impact of coordinate optimization
which dramatically
en
3
−1
−1
within 0.1. ν4
0.1
cm . (199.8)
4289.3(2.5)4089.5 4089.5
hancesaccuracyof
VSCF.
-1 theν4cm c within
4289.3(199.8)4087.0 4087.0(2.5)
hancesaccuracy
of
VSCF.
QQ
Q-112
Q1 Q
Q
Q48Q
Q
: 3783cm:Q3939cm
c Mean
1Q
2
59Q
610
7Q
8Q
9 Q10 Q11 Q12
2 Q3 Q
4the
5
6Q37Q
11
MP2/cc-­‐pVDZ
レベルで
次テイラー展開ポテンシ
absolute
deviation.
Meanc4absolutedeviation.
c MAD
117.4
1.5
MAD
117.4
1.5
Q̃3 Q̃4 Q̃5 Q̃6 Q̃7 Q̃8 Q̃9 Q̃10 Q̃11 Q̃12
ャル関数を生成し，振動状態計算に用いた．図1に
a Optimized for the ground state. The fundamentals
Q̃3 Q̃4normal
Q̃5 Q̃6 Q̃7andQ̃8 Q̃optimized
4 shows
coordinates ofa Optimized
9 Q̃10 Q̃11 Q̃12
thefor the ground state. The fundamentals are the are Figure
B. dimer
Water
dimernormal andWatervirtualenergiesof isthemodals.
water
dimer
for
the
modes.
The coordina
norFigure4 intramolecularshows
optimized
virtual energiesof themodals.
(b) B.
The HamiltonianwithQFF
invariantwithrespect
The
Hamiltonian
with
QFF
is
invariantwithrespect
水素結合ドナー水分子のOH
伸縮振動モードを示す．
bb The results
results
onand
normaloptimized
are in agreement
based
onbasednormalareinthatagreement
mal coordinates
be forcharacterized
as the bending,water candimer
the intramolecularmodes. The
to the coordinatetransformation,
so optimized
thatand
VCI
based
on
to Thethecoordinatetransformation,
so
VCI
based
on
−1
−1
withinwithin 0.1
cm 0.1. cm .
基準座標は水分子の対称・逆対称伸縮振動となる．
andsymmetric
stretchingof bencoordinatescanbecharacterized
as12 the
Q74Q
Q
Q96OH
Q10
QQ
c Mean andQ1 Qmal
Q41Qasymmetric
Q
Q
7Q
811
9Q
11 Q
normal candoptimized
coordinatesareequivalent
thenormaloptimized
coordinates
areinequivalent
in the2 Q3and
52Q63Q
85Q
1210 Qmodes
absolute
deviation.
Mean
absolute
deviation.
each watermolecule,
thoughsymmetricOH
OHstretching
stretch- moand symmetric
andtheasymmetricfull VCI limit.full
However,truncated
VCI
does dependon depend
VCI limit.
However,truncated
VCI
does
on
また，対称伸縮振動はアクセプター水分子が
5%ほ
-1 Q̃
-1 Q̃ symmetricing modes
foundbedelocalized
water
OH st
Q̃slightlyQ̃:Q̃each3807
cm:Q̃3918cm
Q̃thoughQ̃Q̃3arethe coordinates,
thus theand
behaviorof behavior
convergence
the andcoordinates,
thus theof convergence
7 Q̃
811
9 Q̃
11 Q̃to12the other3Q̃to6
74 molecule,
85Q̃
96Q̃
10
1210the4 Q̃5 Q̃water
moleculeby∼6%
inweight.
The
optimization
ing
modes
are
found
to
be
slightly
tocoorthe
ど混ざり，非局在化している．一方，最適化座標はshowsandnormalanddelocalized
optimized
may be different.
Infollowing,
wefollowing,onthe
con-respect
shows 4normal
optimized
coordinatesobe thedifferent.
the
focus
on
the respect
con- Figure 4Figure
The
Hamiltonian
with
QFF
is we
invariantwith
The may
Hamiltonian
withInQFF
isfocusinvariantwithhastransformedtheOH
stretchingmodes
to localOH
watermoleculeby∼the
6%intramolecularin weight.
Themodes.
optimiz
water
dimer
for
vergenceof
VCI[m]-(k)withrespect
to
m.
water
dimer
for
the
intramolecularmodes.
The
norvergenceof
VCI[m]-(k)withrespect
to
m.
to
the
coordinatetransformation,
so
that
VCI
based
on
to
the
coordinatetransformation,
so
that
VCI
based
on
図 1.	
  水 2 量体の水素結合ドナー水分子
ドナー水分子の OH 結合へ局在化する．これは OH stretching
coordinates,
as shown
in Fig.
4 (c),aswhile
hasmaltransformed
thecharacterizedOH
stretchingmodes
tocoordinates
canbethe leavbending
coordinatescanbecharacterized
as locathe
All theelectronic
structure
calculations
werecarriedAll optimized
theelectronic
structure
were
normalandcoordinates
arecalculations
equivalent
in thecarriednormalandoptimized
coordinates
are equivalent
inmal
the
の OH
伸縮振動モードと調和振動数．
35,36
35,36
ing
the
bending
coordinates
unchanged.
present
re-while
stretching
coordinates,
asshown
in Fig.
4stretching(c),andsymmetric
andasymmetric
OHThestretchingmodeso
伸縮振動の強い非調和性を反映している．	
  andsymmetric
andasymmetricOH
out at fullthe VCI
MP2/cc-pVDZ
levelof theory
using
the
out
at
the
MP2/cc-pVDZ
levelof doestheorythelimit.
However,truncated
VCI
dependon
full
VCI
limit.
However,truncated
VCI
doesusingdependon(a)	
  基
準座標と(b)最適化座標．	
  
37
37
sult
indicates
strongof OH
ingthethebending
coordinates
unchanged.
The
prese
each watermolecule,
thoughthesymmetricOH
stretcheachwateranharmonicity
molecule,
thoughthestretching
symmetricO
Gaussian09
program,andthevibrational
structurecal-structure
Gaussian09
program,andthevibrational
calthe coordinates,andthus
thethusbehaviorof convergence
thecoordinates,
andthebehaviorofconvergence	  表 1 に VCI計算の結果を示す．VCI-­‐(n)は量子数を3838drivesthearecoordinates
be localized
toothesulting
indicates
thestrongof theOH
stretingwhich
modes
are
found
to
be
slightly
delocalized
to
modes
found
toanharmonicity
betoslightly
delocalizedtoculations
bybe
MaVi.
culations
byInMaVi.
maydifferent.
the following,
we focuswe
onfocusthe conmay
be different.
In the following,
onmodes,the contheOHbonds.
modes,
which
drivesthecoordinatestobelocalizwatermoleculeby∼6%
inweight.
The
optimizationwatermoleculeby∼6%
inweight.
The
opvergenceofVCI
VCI[m]-(k)with respect
to m. to m.
n 個まで励起を許した計算である．基準座標と最適化座標による
VCI-­‐(8)は 0.1	  vergenceof
VCI[m]-(k)with respect
theOHcalculations
bonds.
VCI
six-dimension
have
been
hasandtransformed
the OHinstretchingmodes
tomodes
local OH
has
transformedtheOH stretchingto
All theAllelectronic
structure
calculations
were carried
the electronic
structure
calculations
were VSCF
carried計算となる．表1から明らかなように，
cm-­‐1 の精度で一致し，従って，benchmark
35,36
carried
out
with
the
above
coordinates
forshown
VSCF
and VCI
calculations
inthe
six-dimension
havew
stretching
coordinates,asshown asinFig.
4intramolec(c),
while
leav
35,36
stretching
coordinates,
in Fig.
4 (c),
out at IV.
the MP2/cc-pVDZ
level of level
theoryusing theout
atthe MP2/cc-pVDZ
of theoryusing theIV. APPLICATIONSAPPLICATIONS
37
ular modes.
The
ZPE
andthefundamentalareingbending
coordinates
unchanged.
Theoutbending
with
above
coordinates
forpresent
theintram
37 vibrational structure caling thecoordinates
unchanged.
TherepGaussian09
program,and theGaussian09
program,and the vibrational structure
cal-thecarried
基準座標を用いた
VCI-­‐(1)はVCI-­‐(8)と比較し大きな誤差を生じている．特に，OH
逆frequencies
38
sult
indicates
thestronganharmonicityofOH
stretchingpresented
in
Table
II.
The
VCI-(8)results
based
on
the
ular
modes.
The
ZPE
andfundamental
frequenci
38
sult
indicates
thestronganharmonicityofOH
culationsby MaVi.
culationsby MaVi.fluoride
A. Weakly-interactinghydrogenA.
Weakly-interactinghydrogenmodes,presented
which
drivesthecoordinates
toa belocalizedtotwo coordinates
are ingood
agreement
withmaximum
in
Table
II.
The
resultso
対称伸縮振動は誤差が大きく，
ν5 と
ν6 fluoride
modes,
which
drivestheVCI-(8)coordinates
tobased
be lo
−1
the
OH
bonds.
deviation
of
0.1
cm
.
Thus,
VCI-(8)servesasa
reftwocoordinates
are
in
good
agreement
with
a
max
表
1.	
  
V
CI-­‐(1),	
  oc-­‐VCI-­‐(1),	
  および
VCI-­‐(8)による水2
theOHbonds.
の誤差はそれぞれ
54.0,	
  115.0	
  cm-­‐1
であandVCI
calculations
insix-dimension
have been
erence toVSCF
evaluate
theof. VSCF
In order to
test
the developed
and program,deviation
ofaccuracy
0.1 VCI
cm−1
Thus,
VCI-(8)servesas
In orderto testthetheorydeveloped
theory and program,
VSCF
andcalculations
inlower-rank
six-dimension-­‐ and
IV. APPLICATIONS
量体の分子内振動モードに対する基本振動数(cmIV.
APPLICATIONS
carriederenceout
withthe above
coordinates
forVSCFthe intramolec
VCI.HFVSCF
andVCI[1]-(8)withnormalcoordinates
yield
we have first studied
modelsystem,in whichfourin HF
to evaluate
theaccuracy
of
and
lower
we have afirststudieda modelsystem,which four
る．一方，最適化座標を用いた
oc-­‐VCI-­‐
carried
out
with
the
above
coordinates
for
the
in
1)．括弧内に VCI-­‐(8)からの誤差，MAD に絶対平ular modes.
The
ZPETheandZPE
fundamental
frequencies
areular
modes.
andfundamental
frequ
A. Weakly-interactinghydrogenhydrogen
fluoride fluoride(1)は精度が格段に改善し，絶対平均誤A. Weakly-interactingin Table in
II. Table
The VCI-(8)
based
on ba
the
均誤差を示す．	
   presentedpresented
II. The results
VCI-(8)results
two coordinates
are in good
agreementwith a maximum-­‐1
two coordinates
are
in
good
agreement
withaまで減少した．
差は 37.5	
   から 4.6	  cm
deviationdeviationofoc-­‐VCI-­‐(1)	
  0.1 cmof−10.1. Thus,In ordertest tothetestdeveloped
theory andprogram,−1 VCI-(8) serves as a ref
VCI-­‐(8)	
  cm. Thus, VCI-(8) serves
In toorderthe developed
theoryandVCI-­‐(1)	
  program,	  	
  evaluate
the accuracyof VSCF ofandlower-rank
havewefirststudieda modela system,
in whichinfour
HF fourerenceoc-­‐VCI-­‐(1)の CI we
次元は系の大きさに対し
to evaluate
the accuracy
VSCF
and l
have
first studied
model system,
which
HF toerenceν 	
  1603.3	  (6.6)	
  
1602.3	  (5.6)	  
1596.7	
  て線形であるため，大規模な分子系に対ν 	  1645.5	  (5.7)	
  
1644.6	  (4.8)	  
1639.8	  
して非常に有効な方法である．	
  
ν 	
  
3640.4	  (21.1)	
  3621.7	  (2.4)	
  3619.3	
  
	
  
ν 	
  
3715.6	
  (22.7)	
  3695.1	  (2.1)	
  3692.9	  	
  [結論]	
  振動状態計算において，座標系
3836.8	  (54.0)	
  3788.4	  (5.6)	
  3782.8	  ν 	  をどのように選べば良いか，という古くν 	
  3909.6	
  (115.0)	
  3801.5	  (6.9)	
  3794.6	
  からの大きな問題を解決した．VSCF エ
MAD	
  
	
  	
  
37.5	
  
	
  	
  
4.6	
  
	
  	
  
ネルギーを最小化することで，座標は適切に局所化する．詳細は当日発表するが，座標が局所化することにより，ポテンシ
ャルエネルギー曲面はコンパクトな形へ変換され，VSCF 波動関数が VCI 計算の良い出発点となる．そのため，座標を最適化することで，相関レベルの振動状態計算は厳密解へより速い収束を示す．座標を最適化することで，振動状態計算の適用範囲が大きく広がる．	
  	
  [参考文献]	
  	
  
1.	
  	
  M.	
  S.	
  Child	
  and	  L.	
  Halonen,	
  Adv.	
  Chem.	
  Phys.	
  57,	  1	  (1984).	
  
2.	
  	
  B.	
  R.	
  Henry,	
  Acc.	
  Chem.	
  Res.	
  10,	
  207	  (1977).	
  
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