#3. C s #1 desymmetrized to the stable C 1 #1. The C 2 #2, C 2 #3, and C 1 #1 are quite
close in energy (5 kJ/mol). This ion has been characterized as a salt of all the alkali
metal ions (except lithium), and the sodium salt is commonly known as borax.
An alternative structure might exist where two of the hydroxyls connect to the
non-bridgehead boron atoms. Initially, four structures of C 2v symmetry (#5–#8)
were tried. One of these (C 2v #6) was an energy minima at all levels except B3LYP/
6-31G*. For this structure at this level, and for the other three structures, there exists
A 2 and B 1 imaginary modes. This suggests desymmetrization to C 2 #5–#8 and C s
#9–#12, respectively. The C 2 #6 (B3LYP/6-31G* only) and #7 structures are stable.
All other attempts at structures of C 2 symmetry revert to either C 2 #7 or C 2v #6. The
C s #10 (B3LYP/6-31G*) and C s #11 is stable. All other attempts at structures of C s
symmetry revert to either C s #11 or C 2v #6.
3.2.2 [B 4 O 9 ]
6−
Another tetraborate ion, observed in the lithium salt (Table 3), is shown in Fig. 3.
Initially we considered structures of C 2h and C 2v symmetry, of which there are two
each. None of these is an energy minimum. For C 2h #1, there are imaginary modes
of irreducible representation A u (desymmetrization to C 2 ), B g (desymmetrization to
C i ), and, at MP2/6-311+G*, B u (desymmetrization to C s #2). For C 2h #2, there are
A u imaginary modes, and at MP2/6-31+G* and MP2/6-311+G*, B u imaginary
modes (desymmetrization to C s #3). For C 2v #1, there are imaginary modes of
irreducible representation A 2 (desymmetrization to C 2 ) and B 2 (desymmetrization to
C s ), whereas for C 2v #2, there is only an A 2 imaginary mode, giving rise to the
stable C 2 #4. Desymmetrization of C 2h #1 to a C i structure results in ascent in
symmetry to C 2h #2. Desymmetrization of C 2v #1 to a C s structure usually results in
ascent in symmetry to C 2h #2, except for B3LYP/6-311+G* and MP2/6-311+G*,
where the C s #1 structure remains. None of the three C s structures are energy
minima. The C s #1 structure desymmetrizes to the stable C 1 #1, whereas attempts to
desymmetrize C s #2 and C s #3 result in ascent in symmetry to C 2 #4.
3.3 Pentaborate Species
3.3.1 [B 5 O 6 (OH) 4 ]
−
The pentaborate ion, [B 5 O 6 (OH) 4 ]
−
, observed as a sodium, potassium, rubidium,
and cesium salt (Tables 1, 2, 4, and 5), is given in Fig. 4. Initially, two structures of
D 2d symmetry were tried, and these were indeed minima at most levels. In addition,
other structures formed from rotating hydroxyls by 180 degrees were also minima
at most if not all levels. These are labelled C 2v #1, C 2 #1, C s #1, and C s #2. The only
possible levels where these structures are not minima are MP2/6-31+G* and MP2/
6-311+G*. Structure D 2d #1 has an imaginary E mode at MP2/6-31+G*, whereas
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close in energy (5 kJ/mol). This ion has been characterized as a salt of all the alkali
metal ions (except lithium), and the sodium salt is commonly known as borax.
An alternative structure might exist where two of the hydroxyls connect to the
non-bridgehead boron atoms. Initially, four structures of C 2v symmetry (#5–#8)
were tried. One of these (C 2v #6) was an energy minima at all levels except B3LYP/
6-31G*. For this structure at this level, and for the other three structures, there exists
A 2 and B 1 imaginary modes. This suggests desymmetrization to C 2 #5–#8 and C s
#9–#12, respectively. The C 2 #6 (B3LYP/6-31G* only) and #7 structures are stable.
All other attempts at structures of C 2 symmetry revert to either C 2 #7 or C 2v #6. The
C s #10 (B3LYP/6-31G*) and C s #11 is stable. All other attempts at structures of C s
symmetry revert to either C s #11 or C 2v #6.
3.2.2 [B 4 O 9 ]
6−
Another tetraborate ion, observed in the lithium salt (Table 3), is shown in Fig. 3.
Initially we considered structures of C 2h and C 2v symmetry, of which there are two
each. None of these is an energy minimum. For C 2h #1, there are imaginary modes
of irreducible representation A u (desymmetrization to C 2 ), B g (desymmetrization to
C i ), and, at MP2/6-311+G*, B u (desymmetrization to C s #2). For C 2h #2, there are
A u imaginary modes, and at MP2/6-31+G* and MP2/6-311+G*, B u imaginary
modes (desymmetrization to C s #3). For C 2v #1, there are imaginary modes of
irreducible representation A 2 (desymmetrization to C 2 ) and B 2 (desymmetrization to
C s ), whereas for C 2v #2, there is only an A 2 imaginary mode, giving rise to the
stable C 2 #4. Desymmetrization of C 2h #1 to a C i structure results in ascent in
symmetry to C 2h #2. Desymmetrization of C 2v #1 to a C s structure usually results in
ascent in symmetry to C 2h #2, except for B3LYP/6-311+G* and MP2/6-311+G*,
where the C s #1 structure remains. None of the three C s structures are energy
minima. The C s #1 structure desymmetrizes to the stable C 1 #1, whereas attempts to
desymmetrize C s #2 and C s #3 result in ascent in symmetry to C 2 #4.
3.3 Pentaborate Species
3.3.1 [B 5 O 6 (OH) 4 ]
−
The pentaborate ion, [B 5 O 6 (OH) 4 ]
−
, observed as a sodium, potassium, rubidium,
and cesium salt (Tables 1, 2, 4, and 5), is given in Fig. 4. Initially, two structures of
D 2d symmetry were tried, and these were indeed minima at most levels. In addition,
other structures formed from rotating hydroxyls by 180 degrees were also minima
at most if not all levels. These are labelled C 2v #1, C 2 #1, C s #1, and C s #2. The only
possible levels where these structures are not minima are MP2/6-31+G* and MP2/
6-311+G*. Structure D 2d #1 has an imaginary E mode at MP2/6-31+G*, whereas
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C. C. Pye