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The instability of H3
Volume 87, number 5
9 Apd 1982
CHEMICAL PKYSICS LETTERS
THE ~STABlLiTY
OF H,
Gion CALZAFERRI
An evphclt solution for the orbital stabllczation energy of H3 can be g~vcnwthin a semiempirical molecular-orbital appreach. Correct predxtions of the instabdxy of H3 arc made mdcpendent from a scrcenmg factor. Some pncrahzations for X3 systems xe possible.
We know the potential surface of H3 because very accurate calculations have been carried out [ 11. We
alo know that the simple HMO model leads to the wrong pcedxtion that H3 is stable 123_But we would like to know why this approach is not satisfactory in the present case and what kmd of result is obtained by applying a more sophisticated semiemplrical model. Such a study may lead to a better qualitative understanding of the difference between H3 and the alkah trimers. From e~pe~ent~ [3] as well as - for G3 [41 - from accurate
theoreucal
investigations.
a IScalled the Coulomb integral, \I,~ the resonance IIItegral and S,, the overlap mtegral; rj = 12,23, 13. Without loss of informatton. we fii the numbermg in a way rhat \h 12I is larger than or equal to IItL3I and rhat llr231 is larger than or equal to lkt3l BYspplykg the defirutions
we
that alkah trimers are stable. Smce the energy difference between the Is and the
know
7s orbltals In rhe hydrogen
atom is very large, it is obof the ground state wirhm
vious that for a descnptlon a simple LCAO MO approach we have but to include
IS orbitals. Tlus leads in all geometric situations to the followmg probiem:
=O,
D
x 1 413
(1)
1
q13
x
423
413
x
=o,
12)
X3+aXtb=O,
BYdefinition q13 and qz3 may only take values between 0 and 1 as long as hii and St] have similar dependence on the distance between the nuclei i andj; eq. (6). Therefore,a and b fuifd the condition
I
3
we can write
2
0 009.2614/82]0000-OOOOj5 02.75 0 1982 North-Holland
0~127 + bzf4 < 0 and the solutions of eq. (2) are 443
Volume 87. number 5
X(rr)=Agcos(~0+~,18), n=O,l,z!,
(3)
cos Q = -413413 do = ?[i(l
[31(1 +Q:
+&3)1””
.
and
413 =S13/S,2
>
(6)
and
(;)I/2
GA
.
413 = @131$3)423
+(r& +‘143)]1/2 .
The elgenvalues e(x,,)
e(X,,)=
423 = s231stz
which shows clearly that qz3 and q13 cannot be varied mdependently:
Wth
nf3-<+9x
9 April 1982
CHEMICAL PHYSICS LETI-ERS
All the mentioned appro.ximations formula for homonuclear systems 11,)= K&Sii ,
-= 9 0‘-’
@a)
where K is an empirical constant.
are given by
[LY--X(ll)hl&[l-X(n)Spl
lead to the same
For the orbltal energy we get
.
(4)
InspectIon of (3) leads to the results gven in table 1. It is interesting that the HMO condrt~on for D3, corresponds to 4t3 =q2j = 1 and that III the case of a linear arrangement we have to,set q,3 = 1 and q13 = 0
Ei=CY(I -X,-S&)/(1
-X,S12)
-
(7)
By setting OL= aH, the orbital stabdtzation A&, of H3 IS =o,
energy
b tH3 )
which leads to the solutionsX_t’ = ?t/z,Xo = 0, = -21/2. The Iast solutions never appear withm Xl the present approach, since qz3 = 0 corresponds to II, = S,, = 0 and thts means necessardy ht3 = St, = 0 because the largest difference m distanced between 7-3 and 13 is dlZ + d,, G 3dt3 and the decrease Of Sir is approximately exponenual. In a nest step we introduce the assumphon that the distance dependence of h,, is approxunately proportional to the overlap integral:
In order to discuss the stability of H, relative to H, + H. we have to compare these two orbital stabiluatlon energies:
4, = 4, f W,, 3Hii) .
A=AEorb(H3)
(3
The best known example of this type ofappro.ulmatlon 1s the Wolfsberg-Hehnholz approach [5] which IS successiully used in extended Htlckel calculations [6]. Other treatments of this type have been discussed by Longuet-Higgins [7], Ballhausen and Gray [S] and Yeranos [9]. lnserrmg (5) inro the defiiitron equations for qz3 and 4t3 leads to the Interesting resuit Table 1 V&~es of,Y, for the two extreme wu~tlons
444
A=q
1 0 -1
Ql3’923’l 1 1 -2
+
energy of Hz + H is equal to
1 - X,S,$-
1-I KS&
1- x,s,,
1 +SHI
U 2
1 - x,s,z -3olH. 1 - XoS,,K 1
- [A&erb(HZ) + AEorb(H)]e
XoS12K f 1-xos,, 1-
-1
1)
-K)(l
+X0(1 -s&)
+sH,)(l
1
.
titer some algebraic manipulation mg interesting result:
x &&s12-
413=423=0
n= I-+Xt =&cos($l+fn)
The orbital stabdzahon
A=(1
4
n = o-x_, =zI@Js(;d) ,1=2--Xo=Aocos(~Q+$r)
1-x,s, 2 1- X,S,2K
aHSI2 -X,&)(1
we fmd the follow-
-X&)
X1X&(3+SHz)
t 2X,]
.
(9)
This result shows that the sign of A is independent of K, as long as K is larger than 1. Smaller values of K never make sense. This means that K appears just as a scaling factor and does not influence the qualitative predictions. Smce the expression
CHEMICAL
Volume 87, number 5
9
PHYSICS LEt-l-ERS
April 1992
of H,. One of the failures is Ihe unreasonable condition qz3 = 1, q L3 = 0. The other one is that the HMO eigenvalues are equal to
bihy
can only be pontive or zero, the sign of A IS determmed by the relative magnitude of
Instead of [a - X(n)/~t~]/[ I - X(M)S,,].
‘sH~/~,~+Xu(1-~~2)-X,X~s~~(3+~“2) and 2X,. The first of these espresslons pOSitlVe,
since
fween -7-
XI
IS negative
and 2X,
can only be dwsys
and -4. BY inserting the expresuons
lies be-
for Xt
and X0 into the second part of eq. (9) and by applyingthesubstitution#t =~@t$,and&,==f$+~rr we get oHstz%J
But 1 - X(rr)St? cannot be set equal to one since otherwlse the splitting between bondmg and antIbonding
orbit&
AE+ =(+
A=(‘-~(1+~~lr)(l-XxIs~z)(I-xos~2)A~~
is largely underestimated.
- &)/(I
and the destabdizstlon
Au =SH*IS&
+ cos#,
(10)
+ [l -SH,-AoSU(3+SH2)cos~,1$cos~0. Smce A0 alone can influence the sign of A we can restnct the discussion to do. @can only take values between ~TI and R. Therefore It IF,easy to see that ~?r~~@r.
-(i)“’
=z cos$Q < -I
and
It is important to notice that the mterdependence of qz3 and q13 leads to QI, =fr$t$rrand#,,=$g+jn which means that for dlscussmg eq. (LO) we can e.g. start at Q. = z n and #t = 2 K and advance in equal increments to Q. = 3 ITand dt = 2 II. This leads to the conclusion that for any reasonable value ofSIT and $.t, the single situation in which $0 might become negative IScos$u = 0, which corresponds toqt3= qq3 = 0 and therefore A0 = (:)‘/2.
But q13 = qz3 = 0
c&responds to infinite distance between Hz and H. In this situation the value ofS,, 1sequal to that of SHY and therefore A,,= l/AO+cos#,
=($)1/2-cos(;li)=0.
The value of A at infirute distance between H2 and H IS predicted to be equal to zero as it should be. Within the model there is in any case no net orbital stabilization for H, relafive to H, + H. This means that H3 is predlcted to be unstable& any geometric arrangement. It is now obvious why the HMO model does not lead to correct predictions regarding the sta-
This cm
easily be shown by looking at the sphttmg of the bondmg and antlbondmg orbitals in Hz. The orbital stablhzalion energy A&+ of H2 ISwithin the approach (I) equal to
AE-
= (11,~ - Sa)/(l
+S) energy -S)
.
IS
equal to
(1 lb)
Since 5’~~ IS calculated to be 0.6 18 [ 101, this means that [AL? 1is more than four times larger than lAf?1. If rhls result IS reasonable, it should be posstble to set IAE-I + IAf?I approximately equal to the energy EpU=l1.37eV(Il]neededfortheBtX~~XtC$ transition: (12) E;== l4E+l+ I4E-I=‘(1zt2-aS)/(I+S)(I-S). The orbital stablhzation energy LS for H, equal to 5 of the dlssoclation cncrgy Do [II]. Therefore WC get Es” “Da/(1
-S),
(13)
which leads to EpU (talc) = 4.477 eV/( I - 0.6 I 8) =
11.7 eV. Ir u intereshng that cq_ (I 3) &ves m addltlon a reasonable explanation for the comparatively small B t ‘_i - X t Zg’ transition energy observed in the alkal dimers. They have much smaller dlssoclauon cnergles [I I] but only shghtly lower values for the (flSlllS) OVCrhp intCgrd, e.g SNaz = 0.6. We can now make another conclusron. Eq. (I) and therefore eq. (6) does not only apply to H,. They describe linear combmations of three equivalent #ISortxtak in any X, system. Thus means that withm the approach applied It is not possible to describe a stable hornonuclear X, molecule such as LiJ , Na3 or KJ. The mam dlsslmilarity between these systems and H3 lies in our initial condttion that the energy difference between the 1s orbitalsand the 2s.2~ orbltals in the hydrogen atom is very large. mere is no equivdent
Volume 87, number 5
CHEMICAL PHYSICS LETlFRS
condition for the akali trimers. It is also clear that a minimal basis set 15not sufficient to descnbe any stable excited state of H,. Such states have been discoiered recently by Henberg [ 131. Within a LCAO MO approach atonnc orbifals rvllh II > 2 have to be Included.
Acknowledgement
I should Ike 10 thank Professor E. Schumacher for encouragmg dIscussIons and for his support This work is part oigrant No. 4 099-0.76.04 financed by the SWUSNational Saence Foundslion.
References (I] P. Sxgbahn and B. Lm, J. Chcm. Phys. 68 (1978) 2157; D.G. TruhLx and C.J. Horowitz, J. Chem. Phys. 68 (1978) 2166,and references therein.
446
9 April 1982
[2j W. Kuaelnigg, Clnmhrung in die theorekxhe Chemie, Band 2 (Verlag Chemie, Wemhcim, 1978). [3j A. Herrmann. S. Leutwyler, E. Schumacher and L. Waste. Helv. Chim. ACP 61 (1978) 453. A. Herrmann, hl. Hoimann, S. Leutwyler, E. Schumacher and L. W&te,Chem. Phys. Letters 62 (1979) 216. [-I] W.H. Gerber and E. Schumacher, J. Chem. Phys. 69 (1978) 1692. 151 hf. Wolfsberg and L. Helmhok. J. Chem. Phys. 10 (1952) 837. [6] R. Hoffmann, I. Chem. Phys. 39 (1963) 1397. and refcrertces therem. [7j H.C. LonguetHi&nsand M. de V. Roberts. Pk. Roy. Sot. AZ-I (1953) 336. A130 (1955) 110. [Sj CJ. Ballhausen and H.B. Gray, Inorg. Chem 1 (196’1) 111. [91 W A. Ycranos, J. Chem. Phys. 4-l (1966) 2207. [IO] E.N. Svcndsen, J. Chem. Educ. 54 (1977) 355. [ 111 C. Herzberg, Spectra of dntomic molecules (Van Nostrand, Prmceton. 1950). [ 121R.S. Mullken, J. Ctim. Phys.46 (1919)497. [ L3] C. Herzberg, J. Chem. Phys 70 (1979) 4806.
9 Apd 1982
CHEMICAL PKYSICS LETTERS
THE ~STABlLiTY
OF H,
Gion CALZAFERRI
An evphclt solution for the orbital stabllczation energy of H3 can be g~vcnwthin a semiempirical molecular-orbital appreach. Correct predxtions of the instabdxy of H3 arc made mdcpendent from a scrcenmg factor. Some pncrahzations for X3 systems xe possible.
We know the potential surface of H3 because very accurate calculations have been carried out [ 11. We
alo know that the simple HMO model leads to the wrong pcedxtion that H3 is stable 123_But we would like to know why this approach is not satisfactory in the present case and what kmd of result is obtained by applying a more sophisticated semiemplrical model. Such a study may lead to a better qualitative understanding of the difference between H3 and the alkah trimers. From e~pe~ent~ [3] as well as - for G3 [41 - from accurate
theoreucal
investigations.
a IScalled the Coulomb integral, \I,~ the resonance IIItegral and S,, the overlap mtegral; rj = 12,23, 13. Without loss of informatton. we fii the numbermg in a way rhat \h 12I is larger than or equal to IItL3I and rhat llr231 is larger than or equal to lkt3l BYspplykg the defirutions
we
that alkah trimers are stable. Smce the energy difference between the Is and the
know
7s orbltals In rhe hydrogen
atom is very large, it is obof the ground state wirhm
vious that for a descnptlon a simple LCAO MO approach we have but to include
IS orbitals. Tlus leads in all geometric situations to the followmg probiem:
=O,
D
x 1 413
(1)
1
q13
x
423
413
x
=o,
12)
X3+aXtb=O,
BYdefinition q13 and qz3 may only take values between 0 and 1 as long as hii and St] have similar dependence on the distance between the nuclei i andj; eq. (6). Therefore,a and b fuifd the condition
I
3
we can write
2
0 009.2614/82]0000-OOOOj5 02.75 0 1982 North-Holland
0~127 + bzf4 < 0 and the solutions of eq. (2) are 443
Volume 87. number 5
X(rr)=Agcos(~0+~,18), n=O,l,z!,
(3)
cos Q = -413413 do = ?[i(l
[31(1 +Q:
+&3)1””
.
and
413 =S13/S,2
>
(6)
and
(;)I/2
GA
.
413 = @131$3)423
+(r& +‘143)]1/2 .
The elgenvalues e(x,,)
e(X,,)=
423 = s231stz
which shows clearly that qz3 and q13 cannot be varied mdependently:
Wth
nf3-<+9x
9 April 1982
CHEMICAL PHYSICS LETI-ERS
All the mentioned appro.ximations formula for homonuclear systems 11,)= K&Sii ,
-= 9 0‘-’
@a)
where K is an empirical constant.
are given by
[LY--X(ll)hl&[l-X(n)Spl
lead to the same
For the orbltal energy we get
.
(4)
InspectIon of (3) leads to the results gven in table 1. It is interesting that the HMO condrt~on for D3, corresponds to 4t3 =q2j = 1 and that III the case of a linear arrangement we have to,set q,3 = 1 and q13 = 0
Ei=CY(I -X,-S&)/(1
-X,S12)
-
(7)
By setting OL= aH, the orbital stabdtzation A&, of H3 IS =o,
energy
b tH3 )
which leads to the solutionsX_t’ = ?t/z,Xo = 0, = -21/2. The Iast solutions never appear withm Xl the present approach, since qz3 = 0 corresponds to II, = S,, = 0 and thts means necessardy ht3 = St, = 0 because the largest difference m distanced between 7-3 and 13 is dlZ + d,, G 3dt3 and the decrease Of Sir is approximately exponenual. In a nest step we introduce the assumphon that the distance dependence of h,, is approxunately proportional to the overlap integral:
In order to discuss the stability of H, relative to H, + H. we have to compare these two orbital stabiluatlon energies:
4, = 4, f W,, 3Hii) .
A=AEorb(H3)
(3
The best known example of this type ofappro.ulmatlon 1s the Wolfsberg-Hehnholz approach [5] which IS successiully used in extended Htlckel calculations [6]. Other treatments of this type have been discussed by Longuet-Higgins [7], Ballhausen and Gray [S] and Yeranos [9]. lnserrmg (5) inro the defiiitron equations for qz3 and 4t3 leads to the Interesting resuit Table 1 V&~es of,Y, for the two extreme wu~tlons
444
A=q
1 0 -1
Ql3’923’l 1 1 -2
+
energy of Hz + H is equal to
1 - X,S,$-
1-I KS&
1- x,s,,
1 +SHI
U 2
1 - x,s,z -3olH. 1 - XoS,,K 1
- [A&erb(HZ) + AEorb(H)]e
XoS12K f 1-xos,, 1-
-1
1)
-K)(l
+X0(1 -s&)
+sH,)(l
1
.
titer some algebraic manipulation mg interesting result:
x &&s12-
413=423=0
n= I-+Xt =&cos($l+fn)
The orbital stabdzahon
A=(1
4
n = o-x_, =zI@Js(;d) ,1=2--Xo=Aocos(~Q+$r)
1-x,s, 2 1- X,S,2K
aHSI2 -X,&)(1
we fmd the follow-
-X&)
X1X&(3+SHz)
t 2X,]
.
(9)
This result shows that the sign of A is independent of K, as long as K is larger than 1. Smaller values of K never make sense. This means that K appears just as a scaling factor and does not influence the qualitative predictions. Smce the expression
CHEMICAL
Volume 87, number 5
9
PHYSICS LEt-l-ERS
April 1992
of H,. One of the failures is Ihe unreasonable condition qz3 = 1, q L3 = 0. The other one is that the HMO eigenvalues are equal to
bihy
can only be pontive or zero, the sign of A IS determmed by the relative magnitude of
Instead of [a - X(n)/~t~]/[ I - X(M)S,,].
‘sH~/~,~+Xu(1-~~2)-X,X~s~~(3+~“2) and 2X,. The first of these espresslons pOSitlVe,
since
fween -7-
XI
IS negative
and 2X,
can only be dwsys
and -4. BY inserting the expresuons
lies be-
for Xt
and X0 into the second part of eq. (9) and by applyingthesubstitution#t =~@t$,and&,==f$+~rr we get oHstz%J
But 1 - X(rr)St? cannot be set equal to one since otherwlse the splitting between bondmg and antIbonding
orbit&
AE+ =(+
A=(‘-~(1+~~lr)(l-XxIs~z)(I-xos~2)A~~
is largely underestimated.
- &)/(I
and the destabdizstlon
Au =SH*IS&
+ cos#,
(10)
+ [l -SH,-AoSU(3+SH2)cos~,1$cos~0. Smce A0 alone can influence the sign of A we can restnct the discussion to do. @can only take values between ~TI and R. Therefore It IF,easy to see that ~?r~~@r.
-(i)“’
=z cos$Q < -I
and
It is important to notice that the mterdependence of qz3 and q13 leads to QI, =fr$t$rrand#,,=$g+jn which means that for dlscussmg eq. (LO) we can e.g. start at Q. = z n and #t = 2 K and advance in equal increments to Q. = 3 ITand dt = 2 II. This leads to the conclusion that for any reasonable value ofSIT and $.t, the single situation in which $0 might become negative IScos$u = 0, which corresponds toqt3= qq3 = 0 and therefore A0 = (:)‘/2.
But q13 = qz3 = 0
c&responds to infinite distance between Hz and H. In this situation the value ofS,, 1sequal to that of SHY and therefore A,,= l/AO+cos#,
=($)1/2-cos(;li)=0.
The value of A at infirute distance between H2 and H IS predicted to be equal to zero as it should be. Within the model there is in any case no net orbital stabilization for H, relafive to H, + H. This means that H3 is predlcted to be unstable& any geometric arrangement. It is now obvious why the HMO model does not lead to correct predictions regarding the sta-
This cm
easily be shown by looking at the sphttmg of the bondmg and antlbondmg orbitals in Hz. The orbital stablhzalion energy A&+ of H2 ISwithin the approach (I) equal to
AE-
= (11,~ - Sa)/(l
+S) energy -S)
.
IS
equal to
(1 lb)
Since 5’~~ IS calculated to be 0.6 18 [ 101, this means that [AL? 1is more than four times larger than lAf?1. If rhls result IS reasonable, it should be posstble to set IAE-I + IAf?I approximately equal to the energy EpU=l1.37eV(Il]neededfortheBtX~~XtC$ transition: (12) E;== l4E+l+ I4E-I=‘(1zt2-aS)/(I+S)(I-S). The orbital stablhzation energy LS for H, equal to 5 of the dlssoclation cncrgy Do [II]. Therefore WC get Es” “Da/(1
-S),
(13)
which leads to EpU (talc) = 4.477 eV/( I - 0.6 I 8) =
11.7 eV. Ir u intereshng that cq_ (I 3) &ves m addltlon a reasonable explanation for the comparatively small B t ‘_i - X t Zg’ transition energy observed in the alkal dimers. They have much smaller dlssoclauon cnergles [I I] but only shghtly lower values for the (flSlllS) OVCrhp intCgrd, e.g SNaz = 0.6. We can now make another conclusron. Eq. (I) and therefore eq. (6) does not only apply to H,. They describe linear combmations of three equivalent #ISortxtak in any X, system. Thus means that withm the approach applied It is not possible to describe a stable hornonuclear X, molecule such as LiJ , Na3 or KJ. The mam dlsslmilarity between these systems and H3 lies in our initial condttion that the energy difference between the 1s orbitalsand the 2s.2~ orbltals in the hydrogen atom is very large. mere is no equivdent
Volume 87, number 5
CHEMICAL PHYSICS LETlFRS
condition for the akali trimers. It is also clear that a minimal basis set 15not sufficient to descnbe any stable excited state of H,. Such states have been discoiered recently by Henberg [ 131. Within a LCAO MO approach atonnc orbifals rvllh II > 2 have to be Included.
Acknowledgement
I should Ike 10 thank Professor E. Schumacher for encouragmg dIscussIons and for his support This work is part oigrant No. 4 099-0.76.04 financed by the SWUSNational Saence Foundslion.
References (I] P. Sxgbahn and B. Lm, J. Chcm. Phys. 68 (1978) 2157; D.G. TruhLx and C.J. Horowitz, J. Chem. Phys. 68 (1978) 2166,and references therein.
446
9 April 1982
[2j W. Kuaelnigg, Clnmhrung in die theorekxhe Chemie, Band 2 (Verlag Chemie, Wemhcim, 1978). [3j A. Herrmann. S. Leutwyler, E. Schumacher and L. Waste. Helv. Chim. ACP 61 (1978) 453. A. Herrmann, hl. Hoimann, S. Leutwyler, E. Schumacher and L. W&te,Chem. Phys. Letters 62 (1979) 216. [-I] W.H. Gerber and E. Schumacher, J. Chem. Phys. 69 (1978) 1692. 151 hf. Wolfsberg and L. Helmhok. J. Chem. Phys. 10 (1952) 837. [6] R. Hoffmann, I. Chem. Phys. 39 (1963) 1397. and refcrertces therem. [7j H.C. LonguetHi&nsand M. de V. Roberts. Pk. Roy. Sot. AZ-I (1953) 336. A130 (1955) 110. [Sj CJ. Ballhausen and H.B. Gray, Inorg. Chem 1 (196’1) 111. [91 W A. Ycranos, J. Chem. Phys. 4-l (1966) 2207. [IO] E.N. Svcndsen, J. Chem. Educ. 54 (1977) 355. [ 111 C. Herzberg, Spectra of dntomic molecules (Van Nostrand, Prmceton. 1950). [ 121R.S. Mullken, J. Ctim. Phys.46 (1919)497. [ L3] C. Herzberg, J. Chem. Phys 70 (1979) 4806.