frequencies suggested many possible structures of lower symmetry. The two possible D 3 structures (via A 1 ”) coalesced into one, but it was not a minimum. The two
possible C 3h structures (via A 2 ’) coalesced into one, and it too is not a minimum.
While desymmetrization along the E” and E’ modes should lead to a C 2 and C s
structure, respectively, it was thought that it would be advantageous to first
desymmetrize to the corresponding C 2v structures (which might ascend in symmetry
back to the corresponding D 3h structures). Neither of these was a minimum, as
expected, and C 2v #1 did indeed ascend to D 3h #1 at the HF levels. The D 3 and C 3h
forms desymmetrize into the stable C 3 #1 and #2 forms, respectively, along the A 2
and A” modes, respectively. The two possible C 2v structures could desymmetrize
into either two C 2 structures, or into one of four possible C s structures. The two C 2
structures coalesce into a structure that is only stable at MP2/6-31G*. C s #3 (from
C 2v #1) and C s #4 (from C 2v #2) ascend in symmetry to either C 3h or C 2v #1.
Neither of the C s structures are stable. The C 2 structure desymmetrizes into C 1 #1 at
MP2/6-31+G* and MP2/6-311+G*, and ascends in symmetry to C 3 #1 at the HF
and B3LYP levels. The C s structures desymmetrize to the stable C 1 #2 and 3 at
most levels. The C 3 #1 structure is the most stable.
3.1.4 [B 3 O 4 (OH) 4 ]
3−
The structure of the triborate ion [B 3 O 4 (OH) 4 ]
3− is given in Fig. 2. The C 2v
structure has at least three imaginary frequencies of irreducible representation A 2 ,
B 1 and B 2 . These suggest desymmetrization to a C 2 or to two different C s structures,
respectively. None of these are energy minima (except C s #1 at B3LYP/6-31G*)
and all desymmetrize to C 1 structures #1–3, respectively. C 1 #1 is the most stable.
This ion has been characterized crystallographically as the sodium and rubidium
salt (see Tables 1 and 4).
3.1.5 [B 3 O 5 (OH) 2 ]
3−
The structure of the triborate ion [B 3 O 5 (OH) 2 ]
3− is given in Fig. 2. The two C 2v
structures are unstable and all contain an imaginary A 2 mode, leading to two C 2
structures that coalesce into the most stable minimum. The higher energy C 2v #2
structure also contains a B 2 mode. The lower energy C 2v #1 structure also contains
either a B 2 mode (B3LYP/6-31G*) or B 1 mode (MP2/6-31+G* and MP2/6-311
+G*). The C s #1 structure, derived from C 2v #2, is only stable at B3LYP/6-31G*.
C s #2, derived from C 2v #1, coalesces into C s #1. C s #3, also derived from C 2v #1, is
not stable. The C 1 #1 structure, derived from C s #1, coalesces into C 2 , except at the
stable B3LYP/6-31+G* and B3LYP/6-311+G* levels, whereas the C 1 #2 structure,
derived from C s #3, coalesces into C 2 at both levels. This ion has been characterized
as the sodium salt (see Table 1).
122
C. C. Pye
possible C 3h structures (via A 2 ’) coalesced into one, and it too is not a minimum.
While desymmetrization along the E” and E’ modes should lead to a C 2 and C s
structure, respectively, it was thought that it would be advantageous to first
desymmetrize to the corresponding C 2v structures (which might ascend in symmetry
back to the corresponding D 3h structures). Neither of these was a minimum, as
expected, and C 2v #1 did indeed ascend to D 3h #1 at the HF levels. The D 3 and C 3h
forms desymmetrize into the stable C 3 #1 and #2 forms, respectively, along the A 2
and A” modes, respectively. The two possible C 2v structures could desymmetrize
into either two C 2 structures, or into one of four possible C s structures. The two C 2
structures coalesce into a structure that is only stable at MP2/6-31G*. C s #3 (from
C 2v #1) and C s #4 (from C 2v #2) ascend in symmetry to either C 3h or C 2v #1.
Neither of the C s structures are stable. The C 2 structure desymmetrizes into C 1 #1 at
MP2/6-31+G* and MP2/6-311+G*, and ascends in symmetry to C 3 #1 at the HF
and B3LYP levels. The C s structures desymmetrize to the stable C 1 #2 and 3 at
most levels. The C 3 #1 structure is the most stable.
3.1.4 [B 3 O 4 (OH) 4 ]
3−
The structure of the triborate ion [B 3 O 4 (OH) 4 ]
3− is given in Fig. 2. The C 2v
structure has at least three imaginary frequencies of irreducible representation A 2 ,
B 1 and B 2 . These suggest desymmetrization to a C 2 or to two different C s structures,
respectively. None of these are energy minima (except C s #1 at B3LYP/6-31G*)
and all desymmetrize to C 1 structures #1–3, respectively. C 1 #1 is the most stable.
This ion has been characterized crystallographically as the sodium and rubidium
salt (see Tables 1 and 4).
3.1.5 [B 3 O 5 (OH) 2 ]
3−
The structure of the triborate ion [B 3 O 5 (OH) 2 ]
3− is given in Fig. 2. The two C 2v
structures are unstable and all contain an imaginary A 2 mode, leading to two C 2
structures that coalesce into the most stable minimum. The higher energy C 2v #2
structure also contains a B 2 mode. The lower energy C 2v #1 structure also contains
either a B 2 mode (B3LYP/6-31G*) or B 1 mode (MP2/6-31+G* and MP2/6-311
+G*). The C s #1 structure, derived from C 2v #2, is only stable at B3LYP/6-31G*.
C s #2, derived from C 2v #1, coalesces into C s #1. C s #3, also derived from C 2v #1, is
not stable. The C 1 #1 structure, derived from C s #1, coalesces into C 2 , except at the
stable B3LYP/6-31+G* and B3LYP/6-311+G* levels, whereas the C 1 #2 structure,
derived from C s #3, coalesces into C 2 at both levels. This ion has been characterized
as the sodium salt (see Table 1).
122
C. C. Pye