High energy-density materials

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High energy-density materials
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HIGH ENERGY-DENSITY MATERIALS:
THE ROLE OF PREDICTIVE THEORY
By
KENNETH JOHN WILSON

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA

2002




ACKNOWLEDGMENTS
I have relied and benefited from numerous people and many facilities during
the course of my graduate studies. First, I would like to acknowledge my committee
chair, Professor Rodney J. Bartlett, whose expertise and consistent enthusiasm were
very much appreciated throughout my graduate career. I would also like to thank
Professors N. Yngve 6)hrn, Hendrik J. Monkhorst, Alexander Angerhofer, Jeffrey L.
Krause, and the late Michael C. Zerner who kindly agreed to serve on my
Supervisory Committee.
I am very grateful for many current and former members of the Quantum
Theory Project for their interaction and support. Of special note are Dr. Ajith
Perera, Professor John Watts, Professor Marcel Nooijen, Dr. So Hirata, Professor
Stanislaw Kucharski, Mr. Anthony Yau, Dr. Erik Deumens, Mrs. Coralu Clements
and Mrs. Judy Parker. I would also like to express my gratitude to Professor Janet
E. Del Bene whose research group I joined as an undergraduate student at
Youngstown State University. Based on that experience, I decided to pursue a
doctorate in physical chemistry at the University of Florida.
This work was supported by the United States Office of Naval Research under
AASERT Award number N00014-97-1-0755, by the United States Air Force Office
of Scientific Research under grant number F-49620-95-1-0130, and by the National
Science Foundation under grant number 9980015. Computer facilities for many of
the calculations were provided by the Department of Defense Major Shared
Resource Center at the Army Research Laboratory in Aberdeen, Maryland. This
dissertation was typeset with I5TEX 2,. I am grateful for the developers of that
software, particularly Ron Smith for the creation of the ufthesis class.




TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ............................. ii
ABSTRACT . . . . . . . . . . . . . . . . . . v
CHAPTER
1 INTRODUCTION .............................. 1
2 UNRAVELING THE MYSTERIES OF METASTABLE 0*4 6
2.1 Experimental Discussion ....................... 7
2.2 Theoretical Discussion ........................ 11
3 STABILIZATION OF THE PSEUDO-BENZENE N6 RING WITH
O X Y G EN . . . . . . . . . . . . . . . . . 17
3.1 Introduction . . . . . . . . . . . . . . . 17
3.2 Computational Methods ....................... 18
3.3 Results and Discussion ........................ 20
3.4 Atomic Charges ............................ 29
3.5 Resonance Stabilization? ....................... 31
3.6 Enthalpy of Formation and Specific Impulse ............ 33
3.7 Conclusions . . . . . . . . . . . . . . . 34
4 HEATS OF FORMATION FOR THE AZACUBANES AND
NITRO-SUBSTITUTED AZACUBANES ................. 38
4.1 Introduction . . . . . . . . . . . . . . . 38
4.2 M ethods . . . . . . . . . . . . . . . . 43
4.3 Results and Discussion ........................ 44
4.4 Conclusions . . . . . . . . . . . . . . . 59
5 CHOICES FOR THE ORBITAL SPACE .................. 63
5.1 SCF Orbitals . . . . . .. . . . . . . . . 63
5.2 and Virtual Orbitals ................... 66
5.3 Density Functional Theory Virtual Orbitals ............. 67
5.4 Frozen Natural Orbitals . . . . . . . . . . . 68
5.5 Illustrative Examples 69
6 CONCLUSIONS 74




APPENDIX
A COMPUTATIONAL IMPLEMENTATION ................ 75
REFERENCES ..... ............................. 76
BIOGRAPHICAL SKETCH ............................ 86




Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
HIGH ENERGY-DENSITY MATERIALS:
THE ROLE OF PREDICTIVE THEORY
By
Kenneth John Wilson
May 2002
Chair: Rodney J. Bartlett
Major Department: Chemistry
This work explores several highly energetic molecular systems that might be
useful as novel fuels or propellants. These systems include 04, N603, and a series of
azacubanes (C8-,,N,,Hs-,) along with their their nitro-substituted derivatives
(C8-.N.(NO2)8-.). For 04, a cooperative experimental-theoretical study was
performed and a novel, long-lasting, van der Waals complex was spectroscopically
observed and identified. However, another isomer of 04, a covalently-bound D3h
form, seems promising as a highly energetic metastable species and merits further
study. The N603 system was studied in terms of a series of nitrogen rings stablized
by coordinate-covalent bonds with oxygen. Out of all of the nitrogen-ring molecules
considered, N603 had the best separation of adjacent atomic charges. Consequently,
it also had the most kinetic stability with a 62.4 kcal mo1-1 activation energy
toward unimolecular dissociation. The series of azacubanes were investigated as
derivatives of the cubane molecule, an extremely dense, shock-insensitive explosive.
The computed heats of formation for all of the azacubanes were larger than
cubane's, however, none of the azacubanes have been synthesized.




To facilitate predictive calculations on other highly energetic and larger
molecules, several improvements for the virtual orbital space were explored. These
include virtual orbitals generated from a potential of a reduced number of electrons,
virtual generated from density functional potentials, as well as the use of
approximate natural orbitals for the virtual space. Numerical results show that
natural orbitals are one of the better sets and recover most of the correlation energy
in a much smaller space.




CHAPTER 1
INTRODUCTION
Electronic structure theory has emerged as a powerful tool to elucidate
molecular structures, energetics, and properties. Developments in the field, coupled
with improvements in computer technologies, have made it fairly routine to
characterize molecular systems with five or fewer atoms. Typical quantities
predicted are molecular potential energy surfaces including minima and extrema,
energy differences with their associated transition states and activation barriers, and
normal harmonic vibrational frequencies with their infrared and Raman intensities.
The coupled-cluster (CC) framework, with an exponential ansatz, is often the
method of choice for solutions to the electronic Schr~dinger equation, as well as the
prediction of properties [1-6]. With the demise of configuration-interaction (CI)
methods in the 1990s, years of chemical literature show that CC methods are both
accurate and routinely applicable. Current research efforts that add quadruple [7],
pentuple [8], and higher excitations [9] (as well as the efficient construction of
spin-adapted wave functions [10-12]) will allow CC theory to describe even more
chemical processes.
Coupled-cluster methods have made great progress in the field of high
energy-density materials (HEDMs) [13, 14]. Potential HEDMs are characterized by
highly energetic systems that are metastable. It is also desirable, but not required,
that HEDMs be environmentally friendly. One example is liquid hydrogen and
oxygen which is used as a propellant in the space shuttle [15]. Other HEDMs
include monopropellants that are not combined with oxygen and that produce
thrust via unimolecular dissociation to gaseous products.




Coupled-cluster methods have many strengths that make them extremely
useful for the study of HEDMs. The first is their ability to survey many different
types of systems. Provided that a one-particle Gaussian basis set has been
developed for the elements of interest and that relativistic effects are small or
included by pseudo potentials, CC methods should be applicable. In fact, a
theoretical approach facilitates the construction of HEDMs by design. For instance,
atoms can be easily changed within the input and substituent effects deduced.
Furthermore, theoretical studies of environmentally unfriendly fuels, propellants and
explosives (i.e., those that produce metals or toxic products) are safer and can be
less costly than physical experiments. Also, CC methods can be highly accurate in
terms of predicted bond lengths, angles and normal harmonic frequencies, and thus
predictive. Often potential HEDMs exploit novel bonding patterns, or rather
bonding patterns that are not yet known, so highly accurate methods are needed.
Figure 1-1. The computed CCSD(T)/aug-cc-pVTZ structure of tetrahedral N4
(tetrazete).
One example of a molecule envisioned, characterized [16-30], and probably
spectroscopically observed [31] within the HEDM effort, is the tetrahedral N4
molecule, shown in Figure 1-1. Theoretical calculations using CC methods show
that the dissociation energy for N4 into 2 N2 molecules is 183 kcal tool-1 [23] and
the activation energy for this process is 61 kcal tool-1 [27]. Furthermore, if
spin-forbidden radiationless decay is considered, the barrier is still 28.2 kcal
mo1-' [25]. Armed with this information, several experimental groups [31, 32]
attempted to prepare and spectroscopically observe N4. Because of its high
symmetry (Td), one of difficulties is that only one vibrational mode of t2 symmetry




is weakly active in the infrared. In fact, a weak transition was observed at 936.7
cm-1 which compares very well with the CCSD(T)/aug-cc-pVTZ computed value of
936 cm-1 [31]. However, the observed and computed frequencies for the isotopically
substituted 15N4 were somewhat less consistent with values of 900.0 cm-1 and 904.4
cm- respectively. Although three vibrations (e, t2, and a,) are active in its Raman
spectra, a higher concentration (> 80 ppm) in solid N2 is required for detection [32].
Figure 1-2. The computed CCSD(T)/aug-cc-pVTZ structure of pentagonal N -
(pentazole).
Another molecule of interest in the HEDM community is the pentazole ring
shown in Figure 1-2. Although arylpentazoles were prepared over forty years
ago [33-36], the N5- moiety has not yet been isolated. While it is certainly true that
strong electron withdrawing groups would stabilize the five-membered nitrogen ring,
a pure nitrogen anion with a large ionization potential is vitally important for the
synthesis of larger nitrogen molecules. Just recently, Christe et al. reported the
synthesis and characterization of the highly energetic N' molecule [37-39].
Subsequent work explored the prospects of combining N+ with the azide anion
(N3-). These attempts have not been successful and a theoretical study of the
possible N8 products showed little promise [40, 41].
With a much larger ionization potential of 5.3 eV compared to that for azide of
2.5 eV, the pentazole anion offers a more promising route toward novel nitrogen
molecules. First, the pentazole anion is completely aromatic while the related
neutral structure is either weakly bound or completely dissociative [42]. Second,
pentazole's activation barrier for unimolecular dissociation is reasonably high (27.4
kcal mol-' at the CCSD(T) level of theory [43-47]) and slightly larger than the 19.9




kcal mo1-' activation barrier for hydrogen pentazole [46, 48]. This suggests that
pentazole would be kinetically stable if it could be prepared. Third, pentazole might
exist bound to a metal atom and a recent study showed that in plane bidentate
complexes are more strongly bound than metallocene-like or sandwich
complexes [49]. Interesting, Gagliardi and Pyykk5 [50] have proposed a larger N7-
ring with 10 7r electons. Such a system, would be able to donate electrons from the
Shell to the metal.
If CC methods are to provide accurate, predictive and computationally
affordable structures, energetics, and properties, a significant limitation is the
dimension of the molecular orbital (MO) basis.a For example, the number of
floating-point operations per closed-shell iteration of the coupled-cluster method
with inclusion of connected single and double cluster operators (CCSD) is
h2 4 occ vir --4 occN i 1 1
where h is the number of symmetry operations, no,, is the number of occupied
orbitals, and gvir is the number of virtual orbitals. Similarly, the coupled-cluster
method with inclusion of connected single and double cluster operators augmented
by a noniterative inclusion of triple excitations, CCSD(T), scales as
1 (n3 +4 4 A3(12
4- (nc2 xAi nocY' v1-2
Given such a power dependence (which is sometimes referred to as the exponential
scaling wall [53]), it is easy to see that applications quickly become prohibitively
expensive with increasing molecular size.
SOften, the dimension of the MO basis is the same as the primitive atomic orbital
(AO) basis. The need for large AO basis sets with high angular momenta is one of
the consequences of the electronic Coulomb cusp condition [51, 52].




Conversely, the power dependence as a function of the number of orbitals also
displays the potential savings if a subset of the full space is used. In this
dissertation, I show that frozen natural orbitals (FNOs) of a reduced dimension can
be generated such that
N,'vir = xNvir where X < 1 (1-3)
Generation of FNOs in the full space is approximate (i.e., done at a lower level of
theory), and hence does not involve a rate limiting step. Although FNOs are of a
smaller dimension than typical canonical Hartree-Fock virtual orbitals, they still
recover nearly all of the correlation energy. This translates to a savings of x4 per
iteration in the number of operations for CCSD and CCSD(T), and even larger
savings for higher-level calculations. Comparisons among SCF virtual orbitals,
those obtained from effective IV.-, and V potentials where a is the number of
valence electrons, those using density-functional theory (DFT) potentials, and
FNOs will be presented.




CHAPTER 2
UNRAVELING THE MYSTERIES OF METASTABLE O~a
Interest in tetraoxygen molecules dates to a 1924 paper by G. N. Lewis [54];
since then, studies of 02 dimers have enjoyed a long history of investigation [55-57].
Theoretical studies of covalently bound 04 species began with Adamantides' 1980
prediction [58] of a bound cyclic (D2d) form. This stimulated considerable further
theoretical effort as this cyclic 04, nearly 4 eV higher in energy than two 02
molecules, appeared to be a promising candidate for a high energy density
material [59-611. Subsequent theoretical studies also identified a D3h form
analogous to S03 at a somewhat higher energy [62, 63]. Although experimentalists
have long observed evidence of van der Waals' complexes of ground state 02
molecules, no evidence has been found supporting the theoretical predictions of
covalent 04 species. In a recent report from the Suits' laboratory [64], 1+1 resonant
photoionization spectra were reported for an energetic, metastable 04 species
produced in a DC discharge [64]. Intense spectra were observed throughout the
region from 280 to 325 n, implying an initial form of tetraoxygen containing at
least 4 eV internal energy relative to 02 + 02, as well as the existence of a higher
excited state through which the ionization takes place. In the absence of plausible
alternatives accounting for all the observations, an energetic covalent 04 species was
considered the most likely candidate. They presented rotationally resolved
a This chapter consists of parts of an article reprinted with permission from
Darcy S. Peterka, Musahid Ahmed, Arthur Suits, Kenneth J. Wilson, Anatoli
Korkin, Marcel Nooijen, and Rodney J. Bartlett, Journal of Chemical Physics,
volume 110, pages 6095-6098. @1999 American Institute of Physics.




0 0 00
vdW cyclic DAh
Figure 2-1. Various possible isomers of 04.
photoionization spectra and photoelectron spectra. Here, we present the theory
component that provides compelling indirect evidence pointing to the identity of
this species as a novel complex, involving one ground state 02 molecule and one in
the metastable c (1EU) state, but not a covalently bonded 04.
2.1 Experimental Discussion
The experiment [64, 65] consists of passing a pulsed molecular beam of oxygen
through electrodes held at ground and 3 to 5 kV so that the discharge occurs in
the collision region of the beam. The discharge is positively biased when detecting
ions, and negatively biased when detecting electrons to inhibit interference from
corresponding species in the beam. The molecular beam is skimmed before entering
a main chamber wherein it is crossed by an unfocused (for wavelength scans) or
loosely focused (for photoelectron spectra) Nd-YAG pumped dye laser doubled to
yield light tunable around 300 nm with a linewidth on the order of 0.08 cm-. The
laser and molecular beams cross on the axis of a time-of-flight mass spectrometer
with velocity map imaging [66] (VELMI) detector, allowing for several different
kinds of experiments to be performed. Mass-selected resonant ionization scans are
effected by recording the total mass-selected ion yield as a function of laser
wavelength. Total photoelectron signals are similarly obtained by reversing the
potentials and recording the integrated electron signal striking the detector as a
function of laser wavelength. Photoelectron images are recorded on the resonant




32.5

33.0

325

305

320

0 +
4
ILA

0.5
0.0 T
30.5

30.900 30.915 30.930
1 1 -, -' -I Ii
31.5 32.0
frequency (X 103 cm -

31.0

Figure 2-2. Raw 0' photoion yield and total photoelectron yield spectra.
Expanded region of the electron spectrum is shown in the inset.
lines using the VELMI technique, calibrated with Ar* ionization, and converted to
electron kinetic energy using established techniques.
Photoionization spectra recorded for m/e=64, 0' from Suits et al. [67] are
shown for the region from 302 to 325 nm in Figure 2-2. Total photoelectron yield
signals are also shown; these signals are recorded under conditions in which the
discharge bias is reversed and the total discharge power in the electron case is lower
(2.5 Watts vs. 6 Watts). The experimentalists ascribe the differences in the ion and
electron spectra to inherent noise in the higher-power ion scans and the
correspondingly higher temperature of the ion scans, giving more 'hot band'

wavelength (nm)
315 310




21323 D4h 04+
1.24 eV

1A2 C3v 04*
7.38 eV
t
D3h 04
'D 0 5.70 eV
D 04
4.46 eV

2A1 C3v 04*
19.06 eV

04+
< 11. eV
04*
< 7.9 eV
04
< 4.1 eV

4B ig D2h 04+
413, C2h 04+
11.75 eV

03 ('A,) + 0 (ID)
6.77 eV
03 (a 3B2) + O (3p)
4.80 eV
02Xy 3- +02 Y -
0 eV
asymptotes

vdW experimental ',cyclic D3h
Figure 2-3. Relative energies (CCSD/TZ2P) with respect to
(32-) along different ionization paths.

contributions. Under the conditions of the experiment, virtually no ions are
observed other than O+. Inset in Figure 2-2 is an expanded view of the electron
yield spectrum in the long wavelength region near 323 nm. Clearly resolved
rotational spectra are observed, with line spacings on the order of 3.2-3.6 cm-1.
Similar rotational structure is also apparent on some lines in the 306 nm region,
although not as pronounced.




CCSD CCSD(T) rotational constants (B) IP
'A, D2d 4.56 4.16 0.254 0.478 0.478 10.98
2 B2,, D4h ionized state 15.28 14.68 0.260 0.520 0.520
covalent A' D3h 5.80 5.07 0.204 0.408 0.408 12.47
A2 C3, excited statea 7.37 0.208 0.406 0.406
4 A2 C3, ionized state 17.43 17.40 0.194 0.368 0.368
vdW 4 B, C2h ionized state 11.83 11.57 0.149 0.185 0.769
4 Bg D2h ionized state 11.83 11.57 0.149 0.185 0.769
a.) For this species, rotational constants are from the EE-EOM-CCSD geometry.

Table 2-1.

Computed CCSD and CCSD(T) relative energies and rotational constants at optimal geometries for 04 species.
The relative energies, in units of electron Volts, are with respect to 02 X (3E) + 02 X (3E-) and rotational
constants are from CCSD(T) geometries in cm-. Also shown are the lowest vertical ionization potentials for the
two covalently bound forms of 04. The nuclear geometry for these calculations was the optimized CCSD(T)/TZ2P
for the ground state.




2.2 Theoretical Discussion
If one of the covalent species is responsible for these spectra, then two critical
issues are 1) accounting for the observed ionization potential of ,-8 eV from the
metastable state or likely 12 eV or so from two ground state 02 molecules, and 2)
finding a bound excited state ,-4 eV above the metastable species. To this end, we
have performed accurate coupled-cluster (CC) calculations with the ACES II
program system [68]. We use a TZ2P basis [69] of Cartesian Gaussians contracted
as (lls6p3d)/[bs3p2d] except as indicated. Table 2-1 presents computed CCSD and
CCSD(T) [1, 2] energies relative to two ground state 02 molecules for several states
of interest. At the CCSD(T) level, we find two covalent forms: the cyclic (D2d) at
4.16 eV and the pinwheel (D3h) at 5.07 eV. We also obtained detailed structures
and vibrational frequencies (shown in Table 2-2) for these two species and
IP-EOM-CCSD [1, 2] vertical ionization potentials for the two covalent forms. The
lowest INs occur at 10.98 for the cyclic and 12.47 for the pinwheel structures.
However, this is much higher than the energy of two photons at 300 nm (-8 eV),
thus outside the range of the experiment. Also shown in Table 2-1 are the energies
of the ionized van der Waals complexes: our result of 11.57 eV is in excellent
agreement with the 11.67 eV CASSCF value of Lindh and Barnes [70]. Also shown
in Table 2-1 are rotational constants for these species. These are not consistent
with the rotationally resolved spectra in the inset in Figure 2-2. Even accounting
for the nuclear spin symmetry for the D3h species, a line spacing on the order of 6B,
or 1.2 cm-1 is the largest expected.
The other question pertains to the existence of a bound excited state of the
covalent species that is 4 eV above the metastable species and whose vibrational
frequencies were previously estimated [64]. A STEOM-CC [71] calculation in the
POLl basis at the geometry of the D2d ground state shows a weakly allowed E state
at 7.10 eV and a dipole forbidden A2 state at 8.54 eV and three others weakly




Table 2-2. Optimized structures in Angstroms and degrees, normal harmonic vibrational frequencies in cm-1 and their
associated intensities in parenthesis with units of kin mo1-' of various 04 isomers computed within the TZ2P basis.
HE MBPT(2) MBPT(4) CCSD CCSD(T)
R 1.392 1.483 1.498 1.460 1.486
dihedral 20.4 28.8 28.9 26.6 27.9
a, 353 411 401 402 400
'A, D2d e 1107 703 652 816 702(0.0)
b2 1062 779 758 856 798(0.1)
b, 1245 851 797 930 809
a, 1222 895 844 985 897
R 1.876 2.056 2.040 1.975 2.012
b2. 405 329 292
2B12u D4h cation big 1250 975 898
aig 1359 1064 961
b2g 1468 1184 1086
eu 999i 1342 852(0.1)
R 1.240 1.304 1.326 1.290 1.312
e' 596 631 584 585 559(5.4)
1A'I D3h a / 779 666 608 666 609(1.7)
el 1026 1861 1476 1030 988(186.2)
1a, 1035 928 817 902 828
R 1.324 1.330
angle 157.7 155.7
'A2 C3, excited state e 188 126(46)
a, 492 485(4.6)
e 1252 717(9.6)
a, 936 867(0.3)




15 -,
t 2I-g 02
/---02 d 'H, (R)
/ 02 d 'Il9 (R)' ... 02
>-- 0 ('D) +0('D)
~1,2 'Fi. (v)
LU 5 0 (P) + 0 (P)
C u
X ):g-
0.75 1.0 1.25 1,5 1.75 2.0 2.25 2.5 2.75
Ro-o (
Figure 2-4. Relevant potential curves adapted from Reference [73], from calculations
of Reference [74]. The Rydberg and ion curves, duplicated and offset
-0.45 eV (the energy of the 02-0' bond) are shown as dashed lines.
allowed, between 9.2 and 10.0 eV. For the D3h form, the first state is a forbidden A"I
that occurs at 7.24 eV with a strong E' state at 8.92 eV, with the next E' state at
12.93 eV. There are five triplet states in the range of 6 to 9 eV for the D2d form,
with the lowest at 6.26. The triplet states start at 7.07 eV for the D3h isomer, with
an E' state at 7.60. However, despite extensive effort, when optimizing the
geometry for excited states either with CCSD when applicable, or EOM-CCSD
analytical gradient techniques [72] to determine if there were bound excited states,
only one singlet state was found. Neither the energy (shown in Table 2-1) nor the
frequencies (shown in Table 2-2) make a persuasive case for this being the possible
intermediate state in the 1+1 experiment. Taking all these points into
consideration, covalently bound energetic tetraoxygen molecules do not appear
likely to be responsible for the experimental observations.




We now consider electronically excited van der Waals complexes. There are
several metastable states Of 02 that may form long-lived van der Waals complexes
of the correct energy. The relevant potential energy curves for 02 are shown in
Figure 2-4. Complexes involving lower lying metastable singlet states are known,
but have neither sufficient energy nor plausible ionization paths to be responsible
for the experimental observations. The Herzberg states of 02, however, do appear
at the correct energy to form complexes that could give rise to the observed spectra.
One of these, the c ('EU) state, is shown in Figure 2-4 based on calculations of
Saxon and Liu [74], adapted from van der Zande et al [73, 75]. This is the only
species that may possess an allowed optical transition in this wavelength region: an
excited I (lrIg) state exists very near the energy of our probe transition (i.e., about
4 eV above this c state). However, earlier calculations indicated that this I ('rIg)
state is repulsive, adiabatically correlating with two ground state oxygen atoms. In
an illuminating series of experiments [73, 75] studying atomic fragments after charge
transfer from cesium atoms to O+, van der Zande et al. explored the nonadiabatic
dynamics and coupling among several of the curves in Figure 2-4 (and with triplet
curves not shown). Most importantly, they showed that the 1 (111.) and 2 (11Hg)
curves shown in Figure 2-4 do not interact strongly, and may be viewed in a
'diabatic' picture, as shown; the 1 (lrI_,) level is a bound or quasibound state. The I
(111g) <-- c (1EU-) transition of 02 is thus a strongly allowed optical transition in 02
with nearly diagonal Franck-Condon factors in the region of 280-330 nm. It is not
clear that this important implication of the observations of van der Zande et al. had
been recognized.
However, direct ionization of this I (111g) state is not possible in this wavelength
region, since the Franck-Condon factors connecting it with the ground state of the
ion are negligible. In fact, the experiments of van der Zande and coworkers show the
importance of the interactions involving the 1 (111.) valence state and the d ( IHg)




Rydberg state that was initially prepared in their experiments. This shows a path
to ionization in this wavelength region via the valence-Rydberg interactions. These
considerations yield the following scenario for this 1+1 ionization process in 0*4,
indicated by the heavy arrows in Figure 2-4. If we begin with a van der Waals
complex between 02 X (3Eu ) and 02 c ('EU ), a fully allowed electronic transition
localized on the c state molecule takes us to a complex involving the 1 (1I1g) state.
This state can either predissociate to give oxygen atoms (and an 02 molecule),
or couple to the d (1rHg) Rydberg state; or the system can dissociate to two 02
molecules. It is likely that all of these occur, no doubt with a strong dependence on
the initially excited vibrational level. If the Rydberg complex is formed, it can then
ionize easily in this wavelength region, and the ionization will be dominated by
Av = 0 transitions owing to the diagonal Franck-Condon factors between the
Rydberg and the ion. If this picture is accurate for 04, it is surprising that no 0' is
seen; this implies some significant differences for the ionization dynamics in the
complex as opposed to the free 02 molecule. In fact, it is precisely in the nature of
these Rydberg-valence interactions that we can expect a profound impact of the
formation of the van der Waals complex. This is because the Rydberg state will be
greatly stabilized in the complex-nearly to the extent of the 0.45 eV bond in
02-02'. The valence state curves will be little- perturbed in comparison. The
location of the Rydberg and ion curves for the complex are shown as dashed lines in
Figure 2-4. This provides a reasonable explanation for the absence of the 0' in
these experiments despite the likelihood that the number density of free 02 C ('E-)
molecules is much greater than those involved in complexes. The fate of the free 02,
upon excitation to the 1 (lI~g) state, is either predissociation via the 2 (111g) state,
or by the triplet states interacting with the d state.
Many of the experimental results can be satisfactorily accounted for by
invoking this complex. The rotational spacing in the long wavelength region, about




3.5 cm-1, is very near 4B, for the c (1EU) state (B, = 0.9 cm-1). This would be
expected, for example, for a T-shaped complex wherein one of the rotational
constants will resemble that of one of the 02 molecules. The photoelectron spectra,
dominated by single peaks, arise because the ionization takes place from a complex
involving the d (1l,~) Rydberg state so that Av = 0 transitions dominate as
mentioned above. Finally, the absence of 0' is readily explained by the very
different Rydberg-valence interactions in the complex as opposed to the free 02.
This picture also accounts for some unusual spectra reported in a closely related
study by Helm and Walter [76]. Their experiments were similar to the studies of
van der Zande et al., but used charge transfer to 0' rather than O+. They reported
clearly resolved vibrational structure in the 02 product kinetic energy distributions
after charge transfer from cesium, which they reluctantly ascribed to coincident
formation of two 02 molecules in v--29, a rather unlikely process. This was
necessary to account for the vibrational spacing of 800 cm-1 observed in the 02
kinetic energy release distributions. Our alternative interpretation of their results
suggests simply the reverse of the ionization process outlined above: electron
transfer from cesium populates the Rydberg state around 7.6 eV, which then
couples efficiently to the metastable 02 X ('E-)-O2 1 (l11.) complex. We suggest
that the structure in the kinetic energy release distributions of Helm and Walter
simply reflects the vibrational structure in the metastable state. For the Herzberg
states, the vibrational frequencies are all on the order of 800 cm-1; the vibrational
frequency in the I (1l.~) state is likely to be similar.
It is important to note that although these spectra are not associated with
covalently bound, energetic 04 species, they may well be present in the molecular
beam.




CHAPTER 3
STABILIZATION OF THE PSEUDO-BENZENE N6 RING WITH OXYGEN'
3.1 Introduction
One of the goals for the development of new high-energy-density molecules has
been to explore the prospects for making novel polynitrogen compounds. Since the
generation of N2 as a propulsion or explosion product is highly recommended by its
very strong triple bond,
citehedmabstract,lauderdale,ferris,korkin,nreview,polynitrogen,perera have used
predictive quantum chemical techniques to investigate the structure, stability,
spectra, and decomposition paths of experimentally unknown polynitrogen
molecules, as have others [15,25, 77-79]. (See Reference [80] for an extensive survey
of pure nitrogen species, from 2 to 12 atoms, their cations, anions and low-lying
excited states). It is apparent that if -CH groups could be replaced by isovalent -N,
there will be a large increase in the amount of energy stored by virtue of the
standard state of nitrogen, N2, and also in the repulsion of electron lone pairs on
adjacent nitrogens. However, we pay a price for this endothermicity, since to retain
the energy for use as a fuel, we must have sufficiently high activation barriers to
unimolecular decomposition as well, and we have also to address the question of
whether non-radiative transitions can play a role in the molecule's
decomposition [25]. We have considered the molecule N4 in a tetrahedral
arrangement [21, 24, 28], as well as the N8 analog of cubane [24, 81]. In both cases,
aThis chapter consists of an article reproduced with permission from Kenneth J.
Wilson, S. Ajith Perera, Rodney J. Bartlett, and John D. Watts, Journal of
Physical Chemistry A volume 105, 7693-7699. @2001 American Chemical Society.




the high symmetry indicates that any decomposition to N2 would be
Woodward-Hoffman forbidden, suggesting significant barriers to decomposition.
Numerical investigation of the barriers shows that N4's is 62 kcal mo1-' [27, 77]
though its effective barrier is close to 30 (because of a low-lying triplet state [25, 27])
and N8's is 19 [15, 79]. In Chapter 4 of this dissertation, we have also investigated
the other azacubanes where 1-7 of the -CH units have been replaced by -N, and the
nitroazacubanes. More energy per molecule can be gained from single rather than
double-bonded nitrogen linkages, recommending N4 and N8 if they could be
synthesized and stabilized, but at the cost of the stability offered by the
double-bonded -N--N- structures. In addition, to impose some organization of
proposed energetic species, we here consider dimers, trimers, etc. of highly energetic
units such as 02NCN and other units [82]. In this paper, the unit is N20.
Beyond N2 and N3, there is little experimental evidence to date that nitrogen
can form stable homonuclear molecules. Just recently, Christe and coworkers have
reported the synthesis of N+AsF- [37] while N+ [83], N+ [84-86] and the diazidyl
N6 complex [87, 88] have been detected spectroscopically as short-lived species.
Notable in the series of homoleptic polynitrogen systems is the absence of the N6
ring. The N6 ring, analogous to benzene, is not a minimum on the potential energy
surface as a planar, hexagonal ring. Instead, including electron correlation, the D6h
structure is a second-order saddle point with the closest minimum, a non-planar
boat-type D2 structure [89]. This feature might be attributed to the large lone-pair
repulsion in the ring. However, we might expect to reduce some of this repulsion by
forming coordinate covalent bonds, -N---O. In fact, such bonds are found in
explosives [90-93] and suggest some interesting consequences for the formation of
novel polynitrogen rings.




Table 3-1. Computed electronic energies in Hartrees with the 6-31G* basis for the nitrogen ring systems considered in this
work along with their transition states for unimolecular dissociation and the reference systems (N2, 02 and N20).
Symmetry SCF MBPT(2) CCSD CCSD(T) B3LYP
N40 C3U -292.24948 -293.14183 -293.11678 -293.16431 -293.74738
N40 TS CS -293.12807 -293.10008 -293.15494 -293.73555
N402 C2h -367.18793 -368.22286 -368.21021 -368.26495 -369.00953
N402 TS CS -367.08451 -368.11780 -368.13412 -368.94015
N60 CS -401.23971 -402.41983 -402.40314 -402.46850 -403.27751
N60 TS C1 -401.20786 -402.41173 -402.39816 -402.46492 -403.27193
N602 C2, -476.03554 -477.40901 -477.38679 -477.46083 -478.42721
N602 TS C1 -476.00358 -477.37440 -477.36196 -477.43747 -478.40574
N603 D3h -550.82526 -552.39317 -552.36192 -552.44360 -553.56742
N603 TS C1 -550.65752 -552.28431 -552.23772 -552.33575 -553.44754
N2 D~h -108.94395 -109.25528 -109.25579 -109.26852 -109.47195
02 D~h -149.61791 -149.94973 -149.95176 -149.96178 -150.26010
N20 CCv -183.68012 -184.20414 -184.18702 -184.21223 -184.57778




3.2 Computational Methods
All the ab initio calculations in this report were performed with the ACES 11
program system [68] while the density functional calculations were performed with
the Q-CHEM computer program [94]. The well-known hierarchy of ab initio
methods ranging from: Hartree-Fock self-consistent field (SCF), second-order
many-body perturbation theory (MBPT(2)), coupled-cluster singles and doubles
(CCSD) to coupled-cluster singles and doubles with a perturbative inclusion of
triples (CCSD(T)) were employed to show the importance of higher dynamic
correlation. Density functional theory with the empirical Becke three-parameter
Lee, Yang and Parr (B3LYP) exchange correlation functional was also used. Both
the ab initio and DFT calculations were performed in the 6-31G* one-particle basis
set using Cartesian Gaussians [69]. This basis corresponds to a modest 15 functions
per heavy atom, but is largely sufficient for the whole series of molecules ranging up
to nine heavy atoms, studied here. To obtain better energetics, single point energies
were computed with the aug-cc-pVDZ basis set [95] at the optimized 6-31G*
geometries (see Table 3-1). Total atomic charges were computed with the natural
bond orbital (NBO) procedure of Weinhold et al [96]. A spin unrestricted wave
function was used to describe the X (IE9-) state of 02 which showed little evidence
of spin contamination. All transition states were confirmed by the presence of only
one negative eigenvalue in the computed Hessian matrix. All core electrons were
omitted from the correlation procedure, however, the virtual orbitals corresponding
to the core electrons where included in the correlated calculation.
3.3 Results and Discussion
In this section, we begin by checking the reliability of our methods by
calculations on the N20 molecule. Then, we explore the trends in structures and
energetics for its dimers and trimers plus related species as a function of increasing
the number of coordinate-covalent bonds. The first series is based on a




1.092 1.178 SCF
1.172 1.193 MBPT(2)
1.136 1.201 CCSD
1.148 1.205 CCSD(T)
1.134 1.193 B3LYP
1.127 1. 185 exp
Figure 3-1. Computed and experimental structure for N20.
four-membered nitrogen-ring system and contains: N40 and N402. In the second
series, we consider the six-membered nitrogen-ring systems: N60, N602 and N603.
The computed and experimental geometries of N20 are shown in Figure 3-1.
The SCF method slightly underestimates both bond lengths, while all other
methods predict bond lengths longer than their experimental values. The computed
CCSD(T) N-N bond length of 1.148 A and N-0 bond distance of 1.205 A compare
to the experimental values of 1.127 A and 1.185 A, respectively. The error is due in
large part to the use of the small 6-31G* one-particle basis set, as the larger
cc-pVTZ basis gives 1.132 A and 1.189 A. However the predicted B33LYP bond
lengths seem slightly better in the 6-31G* small basis. The computed and
experimental harmonic frequencies are presented in Table 3-2. Here, the CCSD(T)
computed frequencies are slightly better than the B33LYP computed values. The
largest difference is for the second E+ normal mode which CCSD(T) underestimates
by 6 cm-1 and the B3LYP method overestimates by 88 cm-1.
The first member of the N4 series is the N40 system and its computed
lowest-energy structure is shown in Figure 3-2. The four-membered nitrogen ring is
not a minimum for this molecule which instead adopts a C3, structure similar to
that of the tetrahedral isomer of N4. One of the N-N bond lengths is on the order of
a single bond (1.45 A [99]) with its CCSD(T) predicted value of 1.458 A and the
other one that is not equivalent by symmetry has a slightly longer value of 1.536 A.
Other isomers of N40 have been considered including a C2v ring structure with
oxygen as part of the ring [100]. However, the transition state for the unimolecular




Table 3-2. Computed harmonic vibrational frequencies in cm-1 and infrared intensities in parentheses with units of km mo1-'

for the N2, 02 and N20
Symmetry
oo) E
rl

molecules. All computed
SCF MBPT(2)
2758 2175
1998 1410
689 575
1393 1288
2633 2247

values are
CCSD
2411
1650
597
1319
2353

from the 6-31G* basis.
CCSD(T) B3LYP
2341 2457
1578 1658
573(6.8) 603(9.0)
1292(42.4) 1342(49.9)
2276(287.5) 2370(308.9)

experiment [97,98]
2358.57
1580.19
596.3
1298.3
2282.1

1E + N2 (De
rE, 02 (De
Eg + N20 (




58.4
57.9
58.2
58.2
63.1 58.6
64.2
63.5
63.6
62.9 1.434 SCF
1.573 MBPT(2)
1.510 CCSD
1.536 CCSD(T)
1.236 1.510 B3LYP
1.213
1.236 1.370
1.234 1.480
1.222 1.435
1.458
1.448
Figure 3-2. Computed structure for N40.
70.2
73.8
72.2 56.4
1 45.7 73.7 58.0
146.5 5.
147.2 58.3
147.0 58.2
67.1
64.1
63.5
1.223 63.7
1.212
1.219 L486 1.456MBPT(2)
1.204 1.449 1.450 CCSD
1.453 1.609 1.489 CCSD(T)
1.454 1.538 1.456 B3LYP
1.566
1.536
Figure 3-3. Computed transition state structure for N40.
N-N
Figure 3-4. Lewis structure of trans-nitrosyl azide.
dissociation of this C2v molecule was located which indicated a barrier between 1
and 2 kcal mol. A somewhat more stable and experimentally-observed [101]
isomer is the trans-nitrosyl azide shown in Figure 3-4. Prompted by its Raman
characterization, Klap6tke and Schulz considered two dissociation pathways for this
molecule [102]. The first pathway involved conversion into a cis isomer, then a cyclic
form, followed by its dissociation into N2 and linear N20. The highest barrier for
this process computed at the MP2/6-31+G* level was 6.7 kcal mol-'. A transition




Table 3-3. Activation energies in kcal mo1-1 for unimolecular dissociation. Values
not in parentheses are from the aug-cc-pVDZ basis at the 6-31G*
optimized geometries. All values include electronic energy differences as
well as zero-point vibrational energy differences and thermodynamic
corrections for 298 K.
SCF MBPT(2) CCSD CCSD(T) B3LYP
N40 8.6 (9.3) 9.3 (10.0) 5.0 (5.6) 6.4 (-1.0)
N402 60.1 (60.1) 65.0 (65.4) 44.8 (45.6) 41.1 (41.4)
N60 17.9 (18.5) 3.4 (4.0) 1.8 (1.8) 1.1 (1.2) 2.3 (2.8)
N602 18.4 (19.2) 20.4 (21.3) 13.9 (14.5) 13.1 (13.8) 12.0 (12.6)
N603 101.2 (86.3) 64.9 (66.6) 74.1 62.4 71.9 (74.7)
98. I 130.9
987 133 7
98.5 3:1326
8,4 132.9
98.2 12713
1.2114 1355
1.2(5 1.30 135 SCF
1. .92 1 324 1.4 88 M B PT (2)
322 1.467 CCSD
1.3 16 1 .1 CCSD(T)
.494 B3LYP
Figure 3-5. Computed structure for N402.
state for the direct dissociation into N2 and cyclic N20 was also found which
indicated a barrier of 24.2 kcal mo1-1.
The transition state structure for the loss of NO from our C, structure is
shown in Figure 3-3. It has C, symmetry with the major difference that one of the
N-N bonds is broken. It is worth noting that the differences between the MBPT(2),
CCSD, CCSD(T) and B33LYP methods are slightly larger than those for the
structure shown in Figure 3-2, indicating that more sophisticated treatments of
electron correlation are needed to describe transition states. The barriers, shown in
Table 3-3, differ at most by 7 kcal mo1- from our most reliable estimate, the
CCSD(T) value of 5.6 kcal mo1-1. Such a small value suggests that it may be
possible to observe this N40 isomer only as a short-lived species, rather than being
suitable for preparation and handling in bulk quantities.




74.4
80.5
78.8
2.284 81.2
1.896
2,135 1.201
1.802.128 1.194
1.210 1.205SC
1.2441.0
1291161.242 MBPT(2)
10. 1.226 CCSD
101.31.213 B3LYP
Il 0
99.9 I116.9
108.5 1.313 1.580 110.4
105.9 1 .240 1.676 111.6
1.297 103.4 1.725 113.9
1.290 89.0 1.780
95.0
94.9
Figure 3-6. Computed transition state structure for N402.
The addition of a second oxygen atom to the N4 ring leads to the completely
planar N402 structure shown in Figure 3-5. The optimized SCF structure belongs
to the D2h point group while all other methods predict a less symmetric C2h
structure. One N-N bond length is between that of a single (1.45 A) and double
(1.25 A) bond and the other, not equivalent by symmetry, is slightly longer than a
typical single bond. Previously, Manaa and Chabalowski have considered another
cyclic isomer of N402 where the oxygen atoms were members of the ring [103]. This
structure does not allow for oxygen to remove as much charge from the nitrogen
atoms and hence is not as stable kinetically b For the linear N402 isomer, there
have been two reports of its synthesis [104,105].
The addition of the second coordinate covalent bond to the four-membered ring
significantly increases its kinetic stability. The transition state structure for the loss
of N20 is presented in Figure 36. This structure has C, symmetry and two
considerably longer N-N bonds. The large differences between the SCF and CCSD
b The barrier for the unimolecular dissociation of the N402 boat isomer is 11.0
kcal mo1-1 at the CCSD(T)/6-31G* level. Watts, J. D.; Wilson, K. J.; Bartlett, R.
J. unpublished work.




116.4 123.3
118.9 115,4 121.0
120.6 115.4 122.6
119.4 114.8 122.2
119.8 114.9 122.4
120.318.
__ 117.7
118.0
1.182 117:5
1.233 117.2
1.215 1.330 1.291 SCF
1.224 1.369 1.265 136M fr2
1.211 1.368 1.321 1.328 MBPT(2
1.380 1.3021.4 CDT
1.373 1.3141.4C SDT
1.300 1.326 B3LYP
Figure 3-7. Computed structure for N60.
114.0 120.4
1.147 118.8 166
1.712 117.8 1.229 119,1 1662
1.573 117.5 1'203 12 3 159
1.618 16.7 1 2. 1 .598
151 116.1 120 .6 1534
1.814 1 1.0 1-528 116.4
Al 164119.3
119.2
1.183 125.2119,4
1.227 127.0 118.8
1.219 126.9 1.168 SCF
1.224 125.2 1,238 1.432 131.9 1,5 B T2
1.209 125.9 1189 1.451 124.5 1 25MBT )
1,260 111.8 1.492 125.9 1.218 CCSD
1.291 114.1 1.447 ;25.5 1.247 CCSD(T)
1.278 114.3 1.431 126.3 1.233 B3LYP
114.6
114.5
Figure 3-8. Computed transition state structure for N60.
structures demonstrate the importance of electron correlation in describing the
transition state. Furthermore, there is a 14 kcal mo1-' difference in activation
energies with the SCF computed value of 60.1 kcal mo1-1 and the CCSD value of
45.6 kcal mo1-1. Because of large T2 amplitudes, we were unable to converge the
CCSD(T) method on this transition state.
The N60 molecule is the first member of the second series with a six-membered
ring. Its structure is nonplanar with C, symmetry as shown in Figure 3-7. All N-N
bond lengths are greater than that of a double bond (1.25 A), but shorter than a
single bond (1.45 A [99]). One interesting aspect of the frequencies and intensities
in Table 3-6 is the substantial difference in the B3LYP and CCSD(T) intensities.
Particularly, the three a' modes near 1000 cm-' have a different distribution of
intensities, though the total intensity in that symmetry is comparable.




1.267 SCF
1.310 MBPT(2)
1.300 CCSD
1.310 CCSD(T)
1.296 B3LYP

Figure 3-9. Computed structure

for N602.
1.146
1.165
1.168
1.173
1.152
14100
145.6
143.6
146.1
151.4
131.2
132.4 112.4
132.7 114.8

1.294
1.334
1.328
114.3 1.329
110.7 1.302
112.3
111.8
112.1 l w 5

114.7 1.331 1.314 113.4 1.169 SCF
114.8 1.331 1.314 112.4 1.226 MBPT(2)
114.8 1.372 1.361
173 115.5 1.366 1.357 1.206 CCSD
1.235373 1375 1.216 CCSD(T)
1.212 1.373 1.375
1.223 1.359 1.373 1.204 B3LYP
1.211
Figure 3-10. Computed transition state structure for N602*

The transition state for the unimolecular dissociation of N60 is shown in Figure
3-8 which corresponds to loss of N2. Our best estimate of the barrier for this process
is 1.2 kcal mol-1 at the CCSD(T)/aug-cc-pVDZ//CCSD(T)/6-31G* level of theory.




1.319 SCF
1.353 MBPT(2)
1.352 CCSD
1.363 CCSD(T)
1.355 B3LYP

Figure 3-11. Computed structure for N603.
Addition of another oxygen atom to the previous structure results in the planar
N602 system shown in Figure 3-9. This molecule has C2, symmetry and all N-N
bond lengths are between those of a double and single bond. Again, the IR
intensities, shown in Table 3-6 vary greatly between CCSD(T) and B3LYP.
The transition state for unimolecular dissociation is shown in Figure 3-10 and
corresponds to loss of N3. The barrier for this process is 13.8 kcal mo1-1 at the
CCSD(T)/aug-cc-pVDZ//CCSD(T)/6-31G* level. Although this value is somewhat
larger than that for N60, it is still too low to facilitate handling on a bulk scale.
Other transitions might occur to products involving N3 and N30+, the latter
isovalent to N4.
Addition of another oxygen results in the highly symmetric N603 structure
shown in Figure 3-11, the trimer of N20. This molecule has D3h symmetry with all
N-N bond lengths equal to 1.363 A at the CCSD(T) level of theory. This distance is
almost exactly between that of a double and single bond, attesting to benzene-like




1.584
1.622
1.628
1.665 54.1
1.621 56.0
56.0
55.8
56.7

1.404

2.537

74.9 2942
75.2 2.423 1.070
74.0 2.434 1.168
1.182 75.2 2.584 1.126
1.268 73.9 1.2761.145
1.215 6
1.230 148.9 81262 ." 3
106.5 148.9 2 79.8 105.7
101.0 150.8 79.0 109.1
100.6 114.6 75.8 107.2
9 9 .5 1 67 1 1 8 .6 1 0 7 9
102.5 114.0 1089
1.910 7 1
2 0155 SC
91.694 MBPT(2)
1.8 11.8 2.015 SCF
1.943 CCSD
1.918 CCSD(T)
141.4 .160 1.816 B3LYP
1.192 132.4 1.266
1.209 142.0 1.200
1.208 139.9 1.221
1.216 137.7 1.215
1.203
Figure 3-12. Computed transition state structure for N603.
delocalization (though no resonance stabilization as discussed in Section 3.5).
Unlike N60 and N602, N603 has comparable IR intensity patterns between B3LYP
and CCSD(T).
The transition state for unimolecular dissociation is shown in Figure 3-12. Our
best estimate at the CCSD(T)/6-31G* level for the barrier of this process is 62.4
kcal mol-1. Being the trimer of N20, there could be an alternative dissociation path
to 3 N20's, analogous to the concerted triple dissociation of s-tetrazine [106],
however all attempts to locate other transition states along different dissociation
pathways were unsuccessful. Initial indications are that such a N603 species should
be capable of being synthesized.
3.4 Atomic Charges
One property which can be related to the computed activation energies is the
computed atomic charges or rather the arrangement of the atomic charges. These




-0.32 MBPT(2)
-0.24 B3LYP

-0.06 0.51 -0.46 SCF
-0.08 0.50 -0.42 MBPT(2)
-0.07 0.40 -0.32 B3LYP -0.56 0.48
-0.49 0.47
-0.40 0.39 0.03 SCF
0.03 SCF
0.01 MBPT(2) -0.31
0.00 B3LYP -0.21
-0.21
-0.18
-0.36
-0.34 0.51
-0.26 0.49 -0.13 SCF
0.40 -0.12 MBPT(2)
-0.10 0.08 -0.06 B3LYP
-0.09 0.08 0.12 SCF
-0.05 0.02 0.13 MBPT(2)
-0.20 0.03 B3LYP
-0.16
-0.11
0.55 -0.18
-0.33 0.52 -0.16
-0.34 0.41 -0.10
-0.27

-0.33 SCF
-0.34 MBPT(2)
-0.28 B3LYP

-0.19
-0.14

Figure 3-13. Computed NBO natural atomic charges.




were computed with the natural bond orbital (NBO) formalism [107] within the
6-31G* basis and are presented in Figure 3-13. In all structures, oxygen has a large
negative charge. However, in the stable structures (N402 and N603), the nitrogen
ring is composed of alternating charges. For example in N402, the smallest charge
difference on adjacent nitrogens is 0.60 at the B33LYP level of theory. In N60 and
N602, the charge on adjacent nitrogens is not as well separated with the smallest
B33LYP differences being 0.07 and 0.13, respectively. Figure 3-13 shows a perfectly
alternating NBO charge on the six-membered ring of N603 with the smallest charge
difference being 0.56.
3.5 Resonance Stabilization?
Another interesting concept with the proposed highly energetic molecules is
their resonance energy and whether they are resonance stabilized. Early
calculations on planar-hexagonal N6 based on Shaik's quantum-mechanical
resonance energy [108] and Dewar's ir-resonance energy [109] indicated that
hexazine was even more aromatic than benzene. However, Glukhovtsev and
Schleyer later reported homodesmic and hyperhomodesmic reactions for hexazine
which showed a destabilizing resonance energy of 17.6 kcal mo1-1 and 10.4 kcal
mo1-', respectively [110, 111].
Gimarc and Zhao have offered an explanation of nitrogen's destabilizing
resonance energy as opposed to that of carbon's, which is based on average bond
energies [112,113]. In this approximation, the total energy of a molecule is the sum
of its bond energies. Since the average carbon-carbon single bond energy is 83 kcal
mo1-1 which is more than half of the carbon-carbon double bond energy of 144 kcal
mo1-', carbon structures with pairs of single bonds rather than double bonds are
lower in energy. For nitrogen, the situation is reversed. The average
nitrogen-nitrogen single bond energy is 43 kcal mo1-1, which is less than half of the




nitrogen-nitrogen double bond energy of 100 kcal mo1-1. Hence, nitrogen prefers
double bonds rather than pairs of single bonds.
To estimate the amount of resonance energy in the novel N402 and N603
molecules, we have used the following isodesmic reactions, in analogy to N6 + 3
NH3 -+3 H2N-N=NH.
0
N\ N H \ NN
N H2N
0
Figure 3-14. Isodesmic reactions for N402 and N603.
Based on the energies of the isodesmic reactions shown in Figure 3-14, both ring
structures have a destabilizing resonance energy. For N402, the
CCSD(T)/aug-cc-pVDZ//CCSD(T)/6-31G* value is 27.6 kcal mo1-1 and for N603,
13.9 kcal mo1-'. Other quantum mechanical methods predict values which agree
closely with CCSD(T) and they are presented in Table 3-4. We also present the
resonance energy for N6 in the D6h conformation. In fact, the value of -17.6 is very
close to our MBPT(2)/6-31G* value for N603 of -18.6 kcal mol -'. For N402, the
resonance energy is slightly more destabilizing, numerically being -36.4 kcal mo1-1.
3.6 Enthalpy of Formation and Specific Impulse
One useful figure of merit for potential fuels and explosives is the material's
enthalpy of formation, Afff, or energy relative to the elements in their standard
states. It has been shown for molecules that do not contain fluorine, the enthalpy of
formation largely parallels the heat of combustion [115]. In Table 3-5, we present
AHf's for the ring molecules considered in this work and the N20 test system. For




Table 3-4. Resonance energies in kcal mol-1 for isodesmic reactions. Values not in parentheses are from the 6-31G* basis, and
values in parentheses are from the aug-cc-pVDZ basis at the 6-31G* optimized geometries. All values include
electronic energy differences as well as zero-point vibrational energy differences and thermodynamic corrections for
298 K.
SCF MBPT(2) CCSD CCSD(T) B3LYP
N402 -38.5 (-34.4) -36.4 (-32.1) -36.5 (-31.6) -32.3 (-27.6) -35.2 (-32.5)
N603 -32.1 (-26.2) -18.6 (-10.7) -26.5 (-18.1) -22.0 (-13.9) -20.3 (-15.9)
N6(D6h) -17.6a
(CH)6 23.9a
a.) These values were computed with the MBPT(4)/6-31G*//MBPT(2)/6-31G* formalism [110,111].




Table 3-5. Enthalpy of formation (AHf) and specific impulse (1,) values for the
ring species considered in this work. AHf is in units of kcal mo1-' and
IP is in seconds. Values not in parentheses are from the aug-cc-pVDZ
basis at the 6-31G* optimized geometries. All values include electronic
energy differences as well as zero-point vibrational energy differences and
thermodynamic corrections for 298 K.
AHf IP
B3LYP experiment [114] B3LYP experiment'
N20 16.7 (17.3) 19.61 163.2 (166.3) 176.9
N40 205.2 (206.4) 447.3 (448.6)
N402 125.4 (125.0) 316.3 (315.8)
N60 169.1 (169.3) 344.5 (344.7)
N602 160.4 (160.2) 311.5 (311.4)
N603 155.6 (154.7) 287.7 (286.8)
a.) Calculated using the experimental AHf and Equation 3-1.
N20, the AHf computed from B3LYP/aug-cc-pVDZ//B3LYP/6-31G* is 17.3 kcal
mo1-1 which is in excellent agreement with the experimental value of 19.61 kcal
mo1-1 [114]. For propellants, the molecular weight is also important and a
material's potential is best measured by its specific impulse, I~p. The specific
impulse in units of seconds can be approximated with the equation [116]:
2 /AHf (kcalsmo1l-1)
Isp(seconds) = 265 .. ...m 1_ (3-1
where MW is the molecular weight in grams per mol. The IP's for the ring species
considered in this work are presented in Table 3-5. The prospective HEDMs that
are stable with respect to unimolecular dissociation, N402 and N603, have IP's of
315.8 and 286.8 seconds, respectively. Both of these offer an improvement to the
224 seconds Isp for hydrazine, the most frequently used monopropellant. A survey of
possible generalizations of the basic hydrazine molecular structure has been
considered elsewhere [117].




SCF MBPT(2) CCSD CCSD(T) B3LYP
e 522 382 465 430(2.4) 453(2.9)
e 841 558 659 584(6.7) 604(10.0)
N40 CUv a, 834 710 772 730(1.1) 778(2-3)
e 1173 762 932 845(15.2) 884(15.7)
a, 1378 912 1156 1059(1.8) 1149(1.1)
a, 1726 1703 1603 1579(344.7) 1657(347.3)
N40 TS Cs al 732i 683i 599i (38.8) 612i (49. 1)
au 251 224 228 222(2.2) 228(2.1)
bu, 443 372 372 328(4.8) 343(4.2)
a. 575 558 519 486 498
bg 782 656 666 622 664
ag 805 690 714 702 725
N402 C2h a. 894 908 898 870 876
au 931 737 772 704(14.6) 743(19.5)
bu 969 866 853 789(93.6) 813(102.0)
a. 1260 1183 1130 1147 1133
bu 1450 1346 1356 1292(57.8) 1315(74.0)
bu 1931 1795 1726 1673(617.8) 1706(758.0)
ag 2217 1879 1935 1865 1904
N402 TS C, a'I 1063i 1417i 1070i (73.4) 801i (23.5)
al 130 153 114 136(0-5) 148(0.5)
a"l 451 322 260 313(2.9) 318(1.0)
a"l 494 490 358 425(19.1) 493(5.1)
al 690 516 549 514(17.3) 532(19.2)
al 693 669 665 651(0.7) 674(1.0)
all 675 673 574 569(32.2) 573(24.7)
a? 924 752 789 746(4.5) 774(2.6)
N60 C, a"l 793 827 710 692(5.5) 711(7.3)
a7 965 857 859 821(19.1) 835(30.4)
al 1167 1055 1091 1049(12.3) 1066(30.1)
a, 1294 1138 1115 1073(23.7) 1091(2.0)
all 1493 1166 1266 1173(0.0) 1172(l.4)
a" 1589 1262 1340 1257(11.6) 1284(23.6)
al 1715 1299 1440 1340(31.2) 1390(38.4)
al 1835 1717 1672 1609(201.6) 1648(269.0)
N60 TS C1 a 882i 649i 511i 416i (51.5) 448i (52.9)

Table 3-6.

Computed normal harmonic vibrational frequencies and infrared
intensities in parentheses for various nitrogen ring structures within the
6-31G* basis except for the CCSD(T) frequencies of the N603 isomer
which were computed with the larger cc-pVDZ basis set. Frequencies are
in units of cm-', intensities are in km mo1-1.




3.7 Conclusions
In this work, we have quantitatively shown how four and six-membered
nitrogen rings can be stabilized by coordinate covalent bonds to oxygen. Other
potentially interesting coordinate covalent structures would include those to BH3.
Our analysis is based on locating the lowest energy transition state for unimolecular

CCSD(T)
98
141(0.4)
438(4.1)
453(16.0)
486(17.5)
578(1.7)
595(1.9)
616(48.8)
709
714(14.3)
713(2.2)
992(11.8)
1036(30.1)
1193(1.6)
1226(22.3)
1367(133.4)
1565(447.0)
1633(65.6)
446i (94.9)
127
201(3.3)
442(7.6)
552(12.9)
662
602
669(33.7)
701
938
875
1052(41.3)
1283(175.0)
1559(469.5)
1660
420i (146.8)

B33LYP
90
146(0.4)
445(2.9)
458(17.9)
536(13.4)
583(2.8)
606(33.3)
628(13.4)
738
745(18.2)
706(4.2)
986(7.9)
1060(36.3)
1237(14.7)
1200(22.0)
1409(175.8)
1596(522.6)
1672(84.7)
319i (49.0)
128
202(4.5)
444(5.7)
562(14.7)
679
690
695(38.6)
712
943
915
1064(34.7)
1271(218.4)
1572(508.9)
1682
293i (114.2)

SCF
168
194
518
569
663
672
703
811
868
877
861
1092
1249
1443
1484
1664
1812
1909
474i
166
236
512
633
791
822
830
840
1038
1134
1252
1510
1788
1930
638i

MBPT(2)
98
145
439
462
541
591
605
706
714
722
768
1041
1085
1211
1294
1413
1653
1681
541i
131
207
442
575
674
628
670
689
988
957
1092
1364
1645
1687
364i

CCSD
113
149
458
482
489
600
612
643
747
751
754
1010
1088
1259
1295
1451
1628
1702
448i
131
206
456
572
697
648
704
721
956
916
1102
1346
1610
1720
604i

a2
bi
a,
bi
b2
a,
b2
b2
N602 C2v, a2
bi
a,
a,
b2
a,
a,
b2
b2
a,
N602 TS Cs a7
e
a2 "
el
e
al
a2 I
a2 "
N603 D3h e"l
al
a2
el
e7
e7
al,
N603 TS C, a

Table 3-6-continued.

Table 3-6--continued.




dissociation. Although we believe we have done this effectively, we realize that with
large molecules, and hence high-dimensional potential energy surfaces, other
decomposition routes may exist. Similarly-transformed equation-of-motion
(STEOM) calculations [71] of the vertical excitation energies at the singlet
optimized geometries of N402 and N603 show the lowest triplet states to be at 60.4
and 56.1 kcal mo1-', respectively. Since both of the triplets are high in energy,
singlet-triplet crossings should not significantly lower the activation energies. In
light of the stability characteristics of the N20 dimer and trimer, the tetramer of
N20 might be expected to have similar properties. Our preliminary investigations
show that N804 has a S4-type structure. Although the smallest charge difference on
adjacent nitrogens of the tetramer is 0.54 at the B3LYP level of theory, the structure
is highly nonplanar. We have not investigated its transition states for unimolecular
dissociation, and therefore cannot comment on its kinetic stability. All told, N402
and N603 are highly energetic molecules which appear to be stable. They are indeed
worth attempts to synthesize. To facilitate their identification, we present harmonic
infrared vibrational frequencies and their associated intensities in Table 3-6.




CHAPTER 4
HEATS OF FORMATION FOR THE AZACUBANES AND
NITRO-SUBSTITUTED AZACUBANES
4.1 Introduction
The cubane system was first synthesized over 35 years ago by Eaton and
Cole [118]. In light of cubane's immense strain energy (166 kcal mol-1) and large
positive heat of formation (148.7 kcal mol-1), cubane's kinetic stability up to 230'C
is quite unique [119,120]. This is in large part due to its highly symmetric structure
which makes many of the dissociation pathways Woodward-Hoffman forbidden.
Furthermore, appreciable geometry changes are only possible if two C-C bonds are
broken simultaneously. Another feature of cubane's caged structure is its high
density of 1.29 g cm-3 [120].
Based on their untypical structure and properties, cubane and its derivatives
have emerged as outstanding candidates for high-energy density materials [21,24].
In fact, many of the nitrosubstituted cubanes have been prepared including
octanitrocubane [121-124]. Since nitrogen is isoelectronic with -CH, some obvious
derivatives are the azacubanes (CNs-nH) and nitroazacubanes (CnNs-(NO2)n).
Chemical intuition based on the standard state of nitrogen suggests that the
azacubanes will be higher in energy than the cubane parent molecule. However,
intuition does not provide a means for estimating the magnitude of this energy
difference. Since experimental data on the thermochemical properties of azacubanes
is nonexistent (although one potential precursor has been made [125]) a theoretical
investigation of these properties is warranted.
Estimating the relative stabilities of various azacubanes having a fixed number
of nitrogen atoms (n) does not present a difficult theoretical problem since the gross




structural features of the possible isomers are similar, resulting in an approximate
cancellation of errors in the electronic structure calculations. However, the energy
difference between different classes of azacubanes (e.g. between the n=6 and n=4
isomers of CnN8-,,H,) is considerably more difficult to predict. Indeed, the most
significant quantity to those interested in the potential use of azacubanes as fuels is
the molar heat of formation (A Hf) defined by the enthalpy change in the reaction:
nH2 + 8-- N2 + nC -+CnN8-.Hn (4-1)
and a similar equation exists for the nitroazacubanes. Direct evaluation of AHf by
calculation of electronic energies for all species involved together with corrections
for zero-point vibrational energies and temperature effects (with the latter
contributions hereafter referred to as "thermodynamic corrections") is a notoriously
difficult theoretical problem due to the different bonding situations present in the
reactant and product sides of the chemical equation. For example, it has been
demonstrated that calculation of the heat of formation of gaseous ammonia by
direct calculation of NH3, N2, and H2 produces results that can oscillate wildly
with the level of theory and choice of basis set.
However, a procedure recommended by Pople and collaborators over three
decades ago based on the use of isodesmic reactions [126,127] provides a convenient
means for determining heats of formation. An isodesmic reaction is defined as one
in which the types of all chemical bonds are conserved in the course of the reaction.
A trivial example of an isodesmic reaction is the conversion of the n--6 azacubane
with nitrogens separated by the body diagonal to that in which the nitrogens are
separated by a face diagonal. In both structures, there are 6 C- C single bonds, 6
C- N single bonds, and 4 C- H bonds. Interest in isodesmic reactions stems from
the fact that the shortcomings of approximate wavefunctions and properties
calculated from them are intrinsically related to the chemical environment of the




atoms making up a molecule. As a result of the structural similarities present on
both sides of the equation, errors in absolute energies calculated for the species
involved in an isodesmic reaction tend to cancel when the overall energy change for
the reaction is calculated. Ideally, one chooses an isodesmic reaction where AHf for
all species except the one of interest have been accurately established
experimentally. It is then a straightforward matter to calculate the heat of
formation of the unknown species from a combination of the experimental results
and the theoretical enthalpy change for the isodesmic process. Such an approach
has been used with considerable success in the past to calculate heats of formation
for a wide variety of molecules.
Nevertheless, the concept of chemical environment tacitly assumed in the
preceding paragraph is oversimplified. For example, one might use the isodesmic
reaction:
16CH4 + C8H8 --- 12C2H6 (4-2)
together with experimental values for ethane and methane to estimate the heat of
formation of cubane via the relation:
AHf (C8H8) = 12AHf (C2H6) 16AHf (CH4) HR (4-3)
where HR is the calculated enthalpy change for the isodesmic reaction. Note that
while Equation 4-2 conforms to our definition of an isodesmic reaction (12 C- C
bonds and 72 C- H bonds on both sides of the equation), it is clear that the C- C
bonds in ethane are different from those in cubane, which is a highly strained
system. Therefore, one should expect the approximate cancellation of errors in
isodesmic reactions of this type to be less satisfactory than for processes such as:

C3H6 + CH4 ---+ C2H6 + C2H4

(4-4)




Table 4-1. Naming conventions for the azacubanes and nitroazacubanes investigated
in this study. The atomic positions are shown in Figure 4-1. For the
azacubanes, the substituent of the carbon atom is hydrogen (X= H), and
for the nitroazacubanes the substituent is a nitro group (X=NO2).
Abbreviated Name Formula 1 2 3 4 5 6 7 8
(CX)8 C C C C C C C C
1-aza (CX)7N N C C C C C C C
1,3-diaza (trans) (CX)6N2 N C N C C C C C
1,8-diaza (opp) (CX)6N2 N C C C C C C N
1,2-diaza (cis) (CX)6N2 N N C C C C C C
1,3,5-triaza (CX)5N3 N C N C N C C C
1,2,5-triaza (CX)5N3 N N C C N C C C
1,2,3-triaza (CX)5N3 N N N C C C C C
1,3,5,7-tetraaza (CX)4N4 N C N C N C N C
1,2,3,5-tetraaza (CX)4N4 N N N C N C C C
1,2,5,8-tetraaza (CX)4N4 N N C C N C C N
1,2,3,7-tetraaza (CX)4N4 N N N C C C N C
1,2,3,6-tetraaza (CX)4N4 N N N C C N C C
1,2,3,4-tetraaza (CX)4N4 N N N N C C C C
1,2,3,5,7-pentaaza (CX)3N5 N N N C N C N C
1,2,3,5,6-pentaaza (CX)3N5 N N N C N N C C
1,2,3,4,5-pentaaza (CX)3N5 N N N N N C C C
1,2,3,4,5,7-hexaaza (trans) (CX)2N6 N N N N N C N C
1,2,3,5,6,8-hexaaza (opp) (CX)2N6 N N N C N N C N
1,2,3,4,5,6-hexaaza (cis) (CX)2N6 N N N N N N C C
1,2,3,4,5,6,7-septaaza (CX)N7 N N N N N N N C
Ns N N N N N N N N
which could be used to determine the heat of formation of propylene from known
values for methane, ethane, and ethylene. A large volume of experience has
demonstrated that enthalpy changes for isodesmic reactions of the latter variety can
be calculated at even low levels of theory such as the self-consistent field (SCF)
approximation with minimal or split-valence basis sets. However, for reactions such
as Equation 4-2 in which the conserved "bond types" are less similar chemically,
higher levels of theory should be used.
In this chapter, we apply isodesmic reaction strategies to study the absolute
heats of formation for all isomers of the n=8 to n=O azacubane and nitroazacubane




systems. Due to the strained nature of these systems and their unusual bonding
environments, the considerations discussed above suggest that calculations based on
a low-level ab initio approach may not be satisfactory. Hence, we have investigated
the sensitivity of the predicted heats of formation by performing calculations at
levels of sophistication ranging from the simple SCF model to coupled-cluster (CC)
treatments that include effects of triple excitations. The purpose of the present
study is threefold. In addition to providing what we believe to be accurate
predictions for the formation enthalpies of the azacubanes and the nitroazacubanes
beyond what is in the current literature [128-131], the systematic study of the
dependence of AHf as a function of the correlation treatment should provide some
guidelines for investigations of related molecules. Finally, we compare the results
obtained here with those predicted by semiempirical molecular orbital approaches
and heats of formation computed directly from the standard states.
/ 7
2
5 ~[/
4 3
Figure 4-1. Numbering scheme used in this work.
Table 4-2. Model compounds for isodesmic reactions and their experimental heats
of formation in kcal mo1-1 [114].
molecule AHf
CH4 -17.9
CH3NH2 -5.4
CH3NO2 -19.3
C2H6 -20.0
NH3 -11.0
N(CH3)3 -5.7
N2H4 22.8




4.2 Methods
The first step in calculating the heat of formation is to determine an isodesmic
reaction for each molecule. A small number of simple molecules were chosen as
sources of different types of bonds. For example, for a C- C single bond, ethane
was used. For a N- C single bond, methyl amine was used. All of the model
compounds are listed in Table 4-2 along with their experimental heats of formation.
The sole restriction is that only molecules with experimentally known heats of
formation in the ideal gas state may be used. This set of molecules was used to
form balanced chemical reactions where both the individual atoms and the number
of each type of bond were balanced. Isodesmic reactions for the azacubanes and
nitroazacubanes are given in Tables 4-3 and 4-6, respectively. One consequence of
the large stoichiometric coefficients is that small errors in the computed energy for
each model compound are multiplied by large numbers.
In this study, CC calculations were done with the ACES 11 package [68]. The
semi-empirical and DFT calculations were performed with the GAUSSIAN 94
package [132]. The calculations were done using the Dunning double ( (DE) basis
set [133,134] with polarization functions from correlated calculations [135]. The
geometries for all molecules were optimized at the SCF level and are available upon
request to interested parties. Using these geometries, the energies were calculated at
the AM1 [136], MIND03 [137], PM3 [138], BLYP, B3LYP, SCF, MBPT(2), CCSD,
and CCSD(T) [139] levels. Pure spherical harmonics (i.e., 5 d-type functions) were
used and all core electrons were omitted from the correlation procedure. We did not
perform CCSD and CCSD(T) calculations for the series of nitroazacubanes.
Frequency calculations were also done at the SCF level, and the frequencies
obtained were used to verify the existence of a structure with no imaginary
frequencies. Moreover, the computed frequencies were used to calculate the
zero-point vibrational energy as well as the thermodynamic corrections for




Table 4-3. Reaction coefficients for the isodesmic reaction: azacubane + a CH4 + P
NH3 y N2H4 + 6 CH3NH2 + E C2H6
azacubane a P 7 6
(CH)8 0 16 0 0 12
1-aza 2 14 0 3 9
1,3-diaza (trans) 4 12 0 6 6
1,8-diaza (opp) 4 12 0 6 6
1,2-diaza (cis) 4 12 1 4 7
1,3,5-triaza 6 10 0 9 3
1,2,5-triaza 6 10 1 7 4
1,2,3-triaza 6 10 2 5 5
1,3,5,7-tetraaza 8 8 0 12 0
1,2,3,5-tetraaza 8 8 2 8 2
1,2,5,8-tetraaza 8 8 2 8 2
1,2,3,7-tetraaza 8 8 3 6 3
1,2,3,6-tetraaza 8 8 3 6 3
1,2,3,4-tetraaza 8 8 4 4 4
1,2,3,5,7-pentaaza 10 6 3 9 0
1,2,3,5,6-pentaaza 10 6 4 7 1
1,2,3,4,5-pentaaza 10 6 5 5 2
1,2,3,4,5,7-hexaaza (trans) 12 4 6 6 0
1,2,3,5,6,8-hexaaza (opp) 12 4 6 6 0
1,2,3,4,5,6-hexaaza (cis) 12 4 7 4 1
1,2,3,4,5,6,7-heptaaza 14 2 9 3 0
N8 16 0 12 0 0
finite-temperature. Given this data, the heats of formation for each of the
azacubanes and nitroazacubanes were determined.
Heats of formation were also computed directly from the standard states of the
elements. Again, this is an extremely demanding test for a theoretical method as
nearly every bond in the molecule is broken. To correct for the standard state of
carbon, we used the experimental heat of formation of 169.9 kcal mol for the 3P
state [114].
4.3 Results and Discussion
The heats of formation for the series of azacubanes are presented in Table 4-4.
The series is arranged in terms of increasing nitrogen content and for the




MINDO3
277.6
262.9
244.0
248.5
256.1
220.6
237.4
245.4
192.7
222.2
227.4
230.8
235.1
242.3
202.7
220.7
228.1
209.2
215.2
222.1
211.8
209.9

PM3
141.4
170.2
201.2
199.3
210.1
234.5
241.5
250.8
270.4
284.8
284.3
292.5
292.1
300.1
329.2
336.1
342.4
387.8
389.0
393.5
446.5
505.6

BLYP
131.9
150.3
168.2
170.3
184.1
185.5
203.2
216.5
202.3
235.1
237.6
248.0
250.5
264.6
266.0
283.7
297.4
329.5
331.4
345.8
393.1
456.2

B3LYP
142.8
161.2
178.7
181.2
195.4
195.4
214.4
228.2
211.4
246.3
249.1
259.7
262.5
277.0
277.2
295.9
310.1
342.4
344.4
359.3
407.3
471.6

SCF
160.5
178.5
194.8
198.4
213.3
209.7
231.4
245.9
223.7
262.7
266.3
276.8
280.4
295.8
292.6
313.6
328.5
360.2
362.9
378.3
426.8
492.4

MBPT(2)
148.8
163.9
177.6
180.7
195.2
190.0
210.3
224.3
201.7
238.2
241.5
251.7
255.1
270.1
264.5
284.5
299.1
326.7
329.2
344.7
388.4
449.1

CCSD
153.4
168.6
182.5
185.5
199.8
195.4
215.3
229.1
207.6
243.4
246.6
256.8
260.0
275.0
270.2
289.7
304.2
332.4
334.4
350.0
394.1
455.1

CCSD(T)
150.4
165.2
178.7
181.6
195.8
191.2
210.8
224.4
203.0
238.3
241.4
251.5
254.7
269.6
264.4
283.7
298.1
325.5
327.5
343.0
386.2
446.2

exp
148.7 [140], 159 [141]

azacubane
(CH)s
1-aza
1,3-diaza(trans)
1,8-diaza(opp)
1,2-diaza(cis)
1,3,5-triaza
1,2,5-triaza
1,2,3-triaza
1,3,5,7-tetraaza
1,2,3,5-tetraaza
1,2,5,8-tetraaza
1,2,3,7-tetraaza
1,2,3,6-tetraaza
1,2,3,4-tetraaza
1,2,3,5,7-pentaaza
1,2,3,5,6-pentaaza
1,2,3,4,5-pentaaza
1,2,3,4,5,7-hexaaza
1,2,3,5,6,8-hexaaza
1,2,3,4,5,6-hexaaza
1,2,3,4,5,6,7-septaaza
Ns

orma" onl ncab rnok or e azlal-u slivTo.

---

--

--

--

Table ,4-4 I lea of fI t; i b 1 1-1 b U1II .

AM1
233.4
275.5
321.3
319.7
329.6
371.2
377.6
385.9
425.8
438.0
438.0
444.7
444.1
450.4
501.3
506.9
511.5
577.4
578.6
581.4
654.8
734.1




azacubanes with the same chemical formula, increasing N- N bonds. For cubane,
there are two experimental values of its heat of formation. In 1966, a value of 148.7
kcal mol-' was obtained directly from cubane [140]. A more recent value of 159 kcal
mol-1 was estimated indirectly from the heat of combustion of
1,4-bis(methoxycarbonyl)cubane [141]. Our most reliable CCSD(T) estimate of
150.4 kcal mol-' is more consistent with the older, direct measurement. Trends in
the AHf values for different methods are better presented graphically in Figure 4-2.
Here, the AM1 and MINDO3 methods show large variations from all of the other
methods. However, PM3, the other semiempirical method, is closer to the ab initio
methods. Figure 4-3 offers a magnified view, where AH, differences with respect to
CCSD(T) are plotted. The BLYP and B3LYP values show the same trend with
large variations along the series. The MBPT(2) and CCSD results more closely
follow the CCSD(T) values for the range of molecules studied.
Another important trend along the series of azacubanes is the energy change in
the isodesmic reactions. The isodesmic reaction energies for the various methods are
presented in Table 4-5 and graphically in Figure 4-4. In the bond energy additivity
model where the energy of a molecule is approximated by the sum of its bond
energies, these values would be zero as they represent strain, resonance
stabilization, and other effects. For nearly all of the azacubanes, -AEisodesmic is
greater than zero suggesting that they are less stable than their molecular
fragments. However, -AEisodesmic does decrease along the series with increasing
nitrogen content. Beginning with cubane, -AEisodesmic is 104.6 kcal mol-1 from
CCSD(T) and decreases to -2.8 kcal mol-1 for Ns. This trend is consistent with an
earlier investigation which showed that introduction of nitrogen into the cube
stabilizes it by r-conjugation of the lone pairs [142]. Also, since nitrogen prefers
angles slightly less than 109.5 degrees as opposed to carbon, more substituted cubes




800
700
600 ---------
0500
E
400-AM 1
SBLYP
300 LY
-SCF
-~MBPT(2)
200 *--CCSD
7- '7CCSD(T)
100 0
m ,m cc w m m m co mm c o c
Cl CS C N Vh v r- X
L'q ci ci n Vi '
NN Ci C6
Figure 4-2. AHf for the azacubanes.




'-U

20
15
75
E
10-
o 5-
0
-5

BLYP
B3LYP
--MBPT(2)
-NK CCSD

Figure 4-3. AHf relative to CCSD(T) for the azacubanes.

25 1

0~ 0 0 CL
m) C4
N' CqJ Cq ~ q Vi-
- ------ --

-10 1

/




Table 4-5. -AEisodesmic in kcal mo-' for the azacubanes.
azacubane AM1 MINDO3 PM3 BLYP B3LYP SCF MBPT(2) CCSD CCSD(T)
(CH)8 187.6 231.7 95.6 86.1 97.0 114.7 103.0 107.6 104.6
1-aza 199.5 187.0 94.3 74.4 85.2 102.5 87.9 92.6 89.2
1,3-diaza (trans) 215.2 137.9 95.1 62.1 72.6 88.7 71.5 76.4 72.6
1,8-diaza (opp) 213.6 142.4 93.2 64.2 75.1 92.3 74.6 79.4 75.5
1,2-diaza (cis) 210.0 136.4 90.5 64.5 75.8 93.7 75.5 80.2 76.2
1,3,5-triaza 235.0 84.3 98.2 49.3 59.1 73.4 53.8 59.1 55.0
1,2,5-triaza 227.9 87.7 91.7 53.5 64.6 81.6 60.6 65.5 61.0
1,2,3-triaza 222.6 82.1 87.6 53.3 64.9 82.7 61.0 65.9 61.2
1,3,5,7-tetraaza 259.4 26.3 104.0 35.9 45.0 57.3 35.3 41.2 36.7
1,2,3,5-tetraaza 244.6 28.8 91.4 41.7 52.9 69.3 44.8 50.0 44.9
1,2,3,5-tetraaza 244.6 34.0 90.9 44.2 55.7 72.9 48.1 53.2 48.0
1,2,5,8-tetraaza 237.8 23.9 85.6 41.1 52.8 69.9 44.8 49.9 44.6
1,2,3,7-tetraaza 237.2 28.2 85.2 43.6 55.6 73.5 48.2 53.1 47.8
1,2,3,6-tetraaza 230.0 21.9 79.7 44.2 56.6 75.4 49.7 54.6 49.2
1,2,3,5,7-pentaaza 264.3 -34.3 92.1 29.0 40.2 55.6 27.4 33.2 27.4
1,2,3,5,6-pentaaza 256.4 -29.9 85.6 33.1 45.4 63.0 34.0 39.2 33.2
1,2,3,4,5-pentaaza 247.4 -35.9 78.3 33.3 46.0 64.4 35.0 40.2 34.0
1,2,3,4,5,7-hexaaza (trans) 269.7 -98.5 80.2 21.8 34.7 52.5 19.0 24.7 17.8
1,2,3,5,6,8-hexaaza (opp) 270.9 -92.5 81.3 23.6 36.7 55.2 21.5 26.8 19.8
1,2,3,4,5,6-hexaaza (cis) 260.2 -99.0 72.3 24.6 38.1 57.1 23.5 28.8 21.8
1,2,3,4,5,6,7-septaaza 276.4 -166.6 68.2 14.8 29.0 48.4 10.1 15.7 7.8
Ns 285.1 -239.1 56.6 7.2 22.6 43.4 0.1 6.1 -2.8




120-
100 X ---
8 0 ------ -
E 60 -~PM3
Ix ~BLYP
B3LYP
E
0 40 ......
24
20 - ---- ... .X .
co N zU Z( Z 9
- 2 I qC ~ ( ) Ci ( D) ( D
Figurel 4l i. 4Esdsi for th aacba




51
have lower strain energies. On this scale, the PM3 values show large variations with
respect to all other methods.
Figure 4-5 presents the Eisodesmi, trends relative to CCSD(T). Again, the
BLYP and B3LYP results show large variations over the series, while the MBPT(2)
and CCSD results are more consistent. The range of the BLYP and B3LYP values
are 27.4 and 33, respectively, while the MBPT(2) and CCSD methods have a much
smaller range of 4.5 and 7 kcal mol-', respectively. The largest difference for the
DFT methods is for N8, where B3LYP differs from the CCSD(T) result by 25.4 kcal
mol '.
Table 4-6. Reaction coefficients for the isodesmic reaction: nitroazacubane + a
CH4 + P NH3 -> y N2H4 + 6 CH3NO2 + c C2H6 + 5 N(CH3)3
nitroazacubane Ce 0 6 E
(CN02)8 24 0 0 8 12 0
I-aza 21 0 0 7 9 1
1,3-diaza (trans) 18 1 6 7 4
1,8-diaza (opp) 18 1 6 7
1,2-diaza (cis) 18 1 6 7
3 3
1,3,5-triaza 15 0 0 5 3 3
1,2,5-triaza 15 1 5 4
1,2,3-triaza 15 2 5 5 R
33
1,3,5,7-tetraaza 12 0 0 4 0 4
1,2,3,5-tetraaza 12 8 2 4 2 8
3 3
1,2,5,8-tetraaza 12 4 3 4 3 2
1,2,3,7-tetraaza 12 8 2 4 2 8
3 3
1,2,3,6-tetraaza 12 4 3 4 3 2
1,2,3,4-tetraaza 12 6 44 4 4
3 3
1,2,3,5,7-pentaaza 9 4 3 3 0 3
1,2,3,5,6-pentaaza 9 6 43 1 7
1,2,3,4,5-pentaaza 9 A0 3 2 R
3 3
1,2,3,4,5,7-hexaaza (trans) 6 8 6 2 0 2
1,2,3,5,6,8-hexaaza (opp) 6 8 6 2 0 2
1,2,3,4,5,6-hexaaza (cis) 6 8 7 2 1
3 3
1,2,3,4,5,6,7-septaaza 3 12 9 1 0 1

0 16 12 0 0 0

0 16 12 0

0 0




25
0
E
. 15
E
.~10
0
(n BLYP
5 ~B3LYP
R MBPT(2)
0 CCSD
'0 10 cc CIO CIO a I ) 4 I
di cq N Q) CD CD (DXci _
PL LI -0 t i 4A CL -r *
-5 cL6 vl L6 w ~ ui
(6 Cq (4 N LQ -Q (R (0
0~ N N (Nc! .
-15
-20

Figure 4-5. -AEj.odesmi, relative to CCSD(T) for the azacubanes.




Table 4-7. Heats of formation in kcal mol-1 for the nitroazacubanes.
nitroazacubane AM1 MINDO3 PM3 BLYP B3LYP SCF MBPT(2)
(CNO2)8 333.2 172.5 165.5 138.7 163.6 230.2 126.6
1-aza 359.1 -32.2 39.6 151.8 175.0 233.0 148.1
1,3-diaza (trans) 396.3 115.7 238.1 170.1 191.2 240.0 147.4
1,8-diaza (opp) 396.0 129.2 246.0 170.1 191.6 241.0 149.1
1,2-diaza (cis) 399.2 -124.2 0.3 185.8 208.4 260.7 187.5
1,3,5-triaza 417.9 -609.4 -377.0 180.4 199.8 239.9 187.9
1,2,5-triaza 428.8 -227.1 -26.4 199.6 220.6 264.7 207.5
1,2,3-triaza 442.6 -189.6 -10.8 220.8 239.0 284.8 222.4
1,3,5,7-tetraaza 452.2 -269.1 47.4 193.1 210.2 241.1 204.1
1,2,3,5-tetraaza 474.7 -187.5 70.0 231.2 250.8 288.4 240.5
1,2,5,8-tetraaza 483.6 -151.0 71.5 238.2 257.9 296.0 244.5
1,2,3,7-tetraaza 477.1 -183.9 73.9 244.9 265.8 306.6 257.5
1,2,3,6-tetraaza 488.3 -151.4 82.3 247.6 267.5 306.7 254.0
1,2,3,4-tetraaza 497.9 -111.7 93.6 267.4 288.2 330.3 230.5
1,2,3,5,7-pentaaza 523.1 -150.2 164.4 260.1 277.9 308.4 269.0
1,2,3,5,6-pentaaza 533.2 -107.4 174.5 281.6 300.8 334.7 290.5
1,2,3,4,5-pentaaza 544.0 -73.1 183.5 297.1 316.6 352.0 303.5
1,2,3,4,5,7-hexaaza (trans) 593.5 -30.9 280.1 326.6 344.4 373.2 332.1
1,2,3,5,6,8-hexaaza (opp) 593.7 -26.5 281.3 330.6 348.9 378.5 337.4
1,2,3,4,5,6-hexaaza (cis) 602.6 10.2 288.5 345.8 364.2 395.0 350.2
1,2,3,4,5,6,7-septaaza 663.8 89.0 393.9 392.3 409.2 434.6 392.4
Ns 734.1 209.9 505.6 456.2 471.6 492.4 449.1




800
700
600
o 500 m
E BY
-~ B3LYP
-SCF
400 MP 2
300
200 ---
100
Cl N CC Z\ -r IF -r o
LC) C6 C6 C
Cs-in Cq Cs n Vi n V
- C q
Figure 4-6. A~Hf for the nitroazacubanes.




For the nitroazacubanes, the heats of formation are presented in Table 4-7 and
graphically in Figure 4-6. Beginning with octanitrocubane, AHf's progressively
increase along the series with increasing nitrogen content of the cube. When
compared to the azacubanes, the difference in the MBPT(2) and SCF values is
much larger due to the presence of more nitrogen atoms with lone electron pairs.
Differences in AHf relative to MBPT(2) are presented in Figure 4-7. All three of
the models (SCF, BLYP, and B3LYP) display the same trend with large variations
over the series. One interesting aspect in the energies is the orientation and
repulsion of NO2 groups. All structures have been fully optimized and have no
imaginary frequencies at the SCF level of theory. Although there is a low barrier for
rotation of a NO2 group, the repulsion of adjacent nitro groups would involve a
much higher energy. For 1,2,3,4-tetraaza, there is a large uncharacteristic difference
in the methods with respect to MBPT(2), apparently due to the difficulty in
describing the repulsion of adjacent nitro groups.
The isodesmic reaction energies for the nitroazacubanes are presented in Table
4-8 and graphically in Figure 4-8. Again, -Eisodesmic decreases along the series with
increasing nitrogen content of the cube. In fact, the Eisodesmic values for the
nitroazacubanes are very close to those for the azacubanes. Another aspect of the
nitroazacubane energies, however is the orientation and repulsion of the -NO2
groups. There is a small peak in the isodesmic reaction energies for the molecule
1,2,3,7-tetraaza. Although not all of the nitro groups are attached to the same face
of the cube (see Figure 4-1), this structure seems to have a large amount of
repulsion.
Heats of formation were also computed directly from the standard states of the
elements. The values for the azacubanes are presented in Table 4-9 and those for
the nitroazacubanes are presented in Table 4-10. The differences of the direct
values with respect to the most accurate isodesmic method are presented in Figures




120
1 0 0 ----- ....
~80
E
~60
(NI WSCF
BLYP
7- B3LYP
S40
20
00 7 1-I
z c' c N
-'! C' C'! c'L 0. -r- i
Fig 2r We V.? .X~ .eatv t. ......... c6r tC4 tot a'IT




Table 4-8. -AEi,odesmic in kcal mol-'
nitroazacubane
(CN02)8
1-aza
1,3-diaza (trans)
1,8-diaza (opp)
1,2-diaza (cis)
1,3,5-triaza
1,2,5-triaza
1,2,3-triaza
1,3,5,7-tetraaza
1,2,3,5-tetraaza
1,2,5,8-tetraaza
1,2,3,7-tetraaza
1,2,3,6-tetraaza
1,2,3,4-tetraaza
1,2,3,5,7-pentaaza
1,2,3,5,6-pentaaza
1,2,3,4,5-pentaaza
1,2,3,4,5,7-hexaaza (trans)
1,2,3,5,6,8-hexaaza (opp)
1,2,3,4,5,6-hexaaza (cis)
1,2,3,4,5,6,7-septaaza
N8

for the nitroazacubanes.
AMI BLYP
298.6 104.2
304.5 97.2
300.4 74.3
300.2 74.3
303.4 90.0
323.1 85.6
312.9 83.7
305.6 83.7
337.4 78.3
317.5 74.1
305.3 59.9
319.9 87.8
310.0 69.3
298.4 67.9
324.7 61.7
313.6 62.0
303.4 56.4
311.6 44.7
311.8 48.7
299.6 42.7
298.3 26.8
285.1 7.2

B3LYP
129.1
120.4
95.4
95.8
112.6
105.1
104.7
102.0
95.4
93.6
79.6
108.7
89.2
88.7
79.6
81.2
75.9
62.5
67.0
61.1
43.8
22.6

SCF
195.7
178.4
144.2
145.2
164.9
145.2
148.8
147.7
126.2
131.3
117.7
149.5
128.4
130.8
110.0
115.2
111.3
91.2
96.6
91.9
69.2
43.4

MBPT(2)
92.2
93.5
72.8
74.5
91.7
93.2
91.6
85.4
89.3
83.3
66.2
100.4
75.8
73.2
70.6
71.0
62.8
50.2
55.5
47.1
27.0
0.1




250
200
0
!,--X- BLYP
-~ B3LYP
o -+-SCF
7-MBPT2
50

0 O Cu 1u Wu C u C u wC u
ol co 0
C" r, -6 b P Z
6 C6 Ld C6a C6
C6 C C Cq cli "q--L r R q (
Cc cq C' C! c'-j .- 0
Cl -l N C6

Figure 4-8. -AEi,,d,,mi, for the nitroazacubanes.




Table 4-9. Heats of formation in kcal mol-' for the nitroazacubanes computed
directly.
azacubane PM3 BLYP B3LYP MBPT(2)
(CN02)s 205.1 162.7 181.0 181.7
1-aza 203.4 171.6 187.4 193.4
1,3-diaza (trans) 204.0 179.9 193.0 203.7
1,8-diaza (opp) 202.0 182.0 195.6 206.8
1,2-diaza (cis) 215.1 194.4 208.5 220.9
1,3,5-triaza 206.8 187.7 197.8 212.7
1,2,5-triaza 216.0 204.0 215.5 232.7
1,2,3-triaza 227.6 215.9 228.0 246.3
1,3,5,7-tetraaza 212.2 194.9 201.8 220.9
1,2,3,5-tetraaza 231.1 224.9 234.2 256.8
1,2,5,8-tetraaza 230.7 227.3 237.0 260.1
1,2,3,7-tetraaza 241.1 236.4 246.4 270.0
1,2,3,6-tetraaza 240.7 238.8 249.2 273.4
1,2,3,4-tetraaza 251.0 251.6 262.5 288.1
1,2,3,5,7-pentaaza 247.3 244.9 252.0 279.4
1,2,3,5,6-pentaaza 256.5 261.1 269.4 299.1
1,2,3,4,5-pentaaza 265.0 273.4 282.3 313.4
1,2,3,4,5,7-hexaaza (trans) 282.3 294.6 301.4 337.3
1,2,3,5,6,8-hexaaza (opp) 283.4 296.4 303.5 339.7
1,2,3,4,5,6-hexaaza (cis) 290.3 309.4 317.1 354.9
1,2,3,4,5,6,7-septaaza 317.3 344.3 350.7 394.6
Ns 352.8 393.6 399.3 451.0
4-9 and 4-10 for the azacubanes and nitroazacubanes, respectively. All methods
vary over a large range and the accurate description of AHf directly seems to be
beyond the current limits of ab initio correlated methods. Thus, calculation of heats
of formation directly from the elements is not the preferable route.
4.4 Conclusions
Although a caged structure might be difficult to achieve synthetically,
molecules of this type have some of the highest densities. Octanitrocubane's density
has been estimated as 1.985 g cm-3 [143]. All of the azacubanes and the
nitroazacubanes are highly energetic molecules. With increasing nitrogen content,
greater heats of formation are achieved as well as more stable structures with




I

respect to fragments of the isodesmic reaction. Semi-empirical methods are
inadequate for this problem, though PM3 is better than the others.
Reparameterized semi-empirical theory as manifested in the transfer Hamiltonian
might be able to overcome these limitations [144].
Although Ns is too unstable for production in bulk quantities, molecules within
the azacubane and nitroazacubane series might offer more kinetic stability, albeit a
lower heat of formation. Some estimates of decomposition pathways and energy
barriers for the azacubanes have been made [15,145].

Table 4-10. -AEisojdesmic in kcal mol-1
nitroazacubane PM3
(CNO2) 172.2
1-aza 178.4
1,3-diaza(trans) 188.2
1,8-diaza(opp) 183.0
1,2-diaza(cis) 196.1
1,3,5-triaza 200.0
1,2,5-triaza 204.3
1,2,3-triaza 215.6
1,3,5,7-tetraaza 215.6
1,2,3,5-tetraaza 229.3
1,2,5,8-tetraaza 226.4
1,2,3,7-tetraaza 233.3
1,2,3,6-tetraaza 237.2
1,2,3,4-tetraaza 244.1
1,2,3,5,7-pentaaza 252.3
1,2,3,5,6-pentaaza 258.0
1,2,3,4,5-pentaaza 262.6
1,2,3,4,5,7-hexaaza(trans) 287.8
1,2,3,5,6,8-hexaaza(opp) 288.9
1,2,3,4,5,6-hexaaza(cis) 291.7
1,2,3,4,5,6,7-septaaza 321.3
N8 352.8

for the nitroazacubanes computed directly.
BLYP B3LYP MBPT(2)
153.5 207.9 168.0
163.0 210.2 184.0
173.7 213.7 199.5
173.7 214.1 201.2
189.4 230.8 218.4
184.5 216.9 213.1
199.7 234.0 233.0
216.8 248.8 248.2
193.7 218.3 223.9
223.6 251.4 260.9
226.6 254.9 265.2
237.3 266.5 277.9
235.9 264.5 274.8
251.6 281.5 293.7
244.9 265.9 284.3
262.3 285.0 306.1
273.8 297.2 319.4
295.6 312.3 342.9
299.6 316.7 348.3
310.7 328.4 361.3
345.5 357.0 398.8
393.7 399.3 450.9




SMBPT2
B3LYP
SBLYP
PM3

azacubane

Figure 4-9. AHf from direct calculations relative to AHf from CCSD(T) isodesmic route.

-80
-100




80
60
40
E
CU 20
0
0o 0 ..U M. -MBPT(2)
04 -20 t 1 --
Ld C6 Ld C
Cl o 6 U
C6 4 4 LY
CL ILi .
,- -60
-8 0 ----------
-100
-120
nitroazacubane
Figure 4-10. AlHf from direct calculations relative to AHf from MBPT(2) isodesmic route.




CHAPTER 5
CHOICES FOR THE ORBITAL SPACE
One of the major limitations for quantum chemical calculations is the
dimension of the molecular orbital basis. In fact, the computational cost for ab
initio correlated methods has a tremendous dependence on the size of the molecular
orbital basis. Despite many chemically-interesting processes, it is prohibitively
expensive to describe systems with more than about 10 first-row atoms or about
300 molecular orbitals using highly correlated methods.
Typically, molecular orbitals are defined from canonical Hartree-Fock (HF)
theory. Although, the HF determinant has the lowest energy as an expectation value
of a single Slater determinant with the Hamiltonian, the excited or 'virtual' orbitals
are purely a by-product of the HF calculation in a basis set,' so they are not
necessarily the best set for correlated calculations. In this chapter, we review the
merits of Hartree-Fock theory, and provide alternate choices for molecular orbitals.
5.1 SCF Orbitals
In Hartree-Fock theory [146], the exact Hamiltonian is approximated by a sum
of one-electron Fock operators
_n
i i .F(i) = h(i) + vef (i) (5-2)
aFor atomic and diatomic systems, it is possible to solve the Hartree-Fock equa-
tions numerically, rather than in a finite set of Slater- or Gaussian-type functions. In
these numerical solutions, virtual orbitals are not defined.




The set of orbitals, {9,}, are optimized to give the lowest electronic energy while
remaining orthonormal. Such conditions are imposed using Lagrange multipliers
and functional derivatives.
E [{(,}] = Eo [{n)] i i((i j) ij) (5-3)
ij
Since L is real and (ilj)* = (jli), the multipliers constitute an Hermitian matrix.
AXi = Aiy (5-4)
Taking the partial derivative of with respect to p* yields:
6C 1 1
= (1Pk hl~k) + E(6Vkljllkj) + ((ViPkll i(k) E Akj(j 4kl lIj)
61Pk 2
(5-5)
Setting Equation 5-5 equal to zero, factoring 6lk from every term, and combining
the summation indices gives:
hI k-k Ilk) K kkj l(j) = 0 (5-6)
3 3
where J and K have been introduced to denote the Coulomb and exchange
operators, respectively. Because vey = -j (j1 Ky), Equation 5-6 is a
pseudo-eigenvalue equation in terms of the Fock operator. It is often beneficial to
diagonalize the A matrix to provide an energy for each orbital.
E = UtAU (5-7)
However, there are several considerations to obtaining orbital energies. The
first is whether expectation values will differ between the two wave functions. If A is
diagonalized by a real unitary matrix (Ut = U-1; UtU = 1), the new wave function
can be written in terms of the old one as

(5-8)

I') = det(U)|W)




and the expectation values as

(5-9)

Hence, all expectation values will be invariant to unitary transformations of the
wave function. The second question is whether the Fock operator is invariant. The
only part of the Fock operator that depends on the orbitals is vef and it is
considered below:

= p' (2) 1 r- P (2)dr2
Sr12
= EUE (p (2)1- P12 Uej oj(2) dT2
f ()k k r12
= 6 (klyg(2) 1 rzPl2 1(2)dTa
kl
= k 1(2) 1 Pk(2)dT2
wri k t12 A (2) e ti
write the SCIF canonical equation as

vefl (1)
Consequently, we can

(5-10)

(5-11)

However, thirdly, the canonical orbitals are not typically localized in a region of
space, but rather spread over many atoms. This complicates the interpretation of
molecular orbitals in terms of chemical bonds which are mostly localized.
One of the features of the canonical orbital energies is that they may be
interpreted as approximate ionization potentials. This is Koopmans' theorem and it
involves the same unrelaxed molecular orbitals for the n-1 system as in the n
electron system. For instance, the energy of an n electron system is

E(n) = ~ll>+ ~jl
i i~2

(5-12)

(f'lfl ') = det (UtU) (9q1A|9) = det (1) (W|I1|W)




Provided the orbitals are frozen, the energy is very similar for the n-1 system.
n-1 n-1
E(n 1) = E(ilhli) + ( j~')(-3
The ionization potential (IP) is the difference between the n-1 and n electron
energies
IP = e(n) E(n 1) = -hn - nj[[lnj) + Y injjin) = -En (5-14)
which is also the orbital energy for whichever electron is removed.
5.2 'n-1 and ifn_,, Virtual Orbitals
With some inspection, it is easy to realize that the character of the occupied
and virtual elements of the Fock matrix are quite different.
n
Fpq = hpq + j:(i*pjiq) (5-15)
i
For the occupied orbitals, the effective potential terms cancel, so they are
determined in the potential of n-1 electrons. However, for the virtual orbitals, p and
q are never equal to i, so they are determined in the potential of n electrons, and
hence, might be more appropriate for the n+1 electron system. Furthermore, since
the HF energy is invariant to any definition of the virtuals, it might be beneficial to
redefine them [147]. For CC or CI methods that include all excitations of single,
double, etc. type, the computed energies and properties will be invariant to virtual
orbital transformations. However, the invariance is lost if only a subset of the
virtuals are retained. Thus, it is possible to define better virtuals where subsets
retain most of the correlation energy, as they require much less time than the
calculations in the full virtual space.
Kelly [148-152] investigated excluding different orbitals from veff to determine
virtual orbitals and MBPT correlation energies for various orders. It is, however,
somewhat ambiguous which occupied orbital should be excluded. A more




satisfactory route in modifying vef f is to exclude each occupied orbital in an
averaged way [153].
nl I n- lef (5-16)
n
It is also possible to envision excluding a larger number of electrons or even
fractional numbers of electrons. There ultimate justification will be that a larger
amount of correlation energy is obtained in a smaller subspace. If this is done in the
same averaged way, the new potential has the form
?n _n a ovefS (5-17)
n
Numerical results for the 'Vn-l and l- potentials where a is the number of valence
electrons are presented in Section 5.5.
5.3 Density Functional Theory Virtual Orbitals
Although originally formulated for solids, today, density functional theory
(DFT) is routinely applied to atoms and molecules [154]. The one-particle
Kohn-Sham equations are very similar in form to the Hartree-Fock equations,
except that the Kohn-Sham effective potential is
vef f(r) = ag[p] + E'c [p' (5-18)
In Kohn-Sham theory, veff contains Coulomb repulsion, exchange, and correlation,
and in principle is exact provided the exchange-correlation functional is correct.
One open question is how DFT virtual orbitals will perform in high-level CC
calculations, since unlike SCF orbitals an electron in an occupied or virtual orbital
feels the same potential. In Section 5.5, CC correlation energies using virtual
orbitals from the LDA/VWN [155,156] and PW91 [157] functionals are presented
and compared with other choices for the virtual space.




5.4 Frozen Natural Orbitals
Natural orbitals were introduced by Dirac [158] and extensively studied by
Lawdin [159]. They are defined as the eigenvectors of the one-particle density
matrix and provide the fastest convergence of the configuration-interaction (CI)
expansion [159,160]. One of the potential problems with natural orbitals is that a
correlated wave function is needed for their determination. (They can also be
determined from orthogonalizing the Feynman-Dyson amplitudes using some
self-energy approximation [161,162]). Several authors, including Meyer [163], used
pseudo-natural orbitals obtained from an approximate density matrix at a lower
level of theory to generate the natural orbitals. Since the density matrix is from a
lower level of theory, the generation of pseudo-natural orbitals does not involve any
rate limiting steps or is not a bottleneck for the calculation.
In practice, it is advantageous if there are no off-diagonal occupied-occupied,
fij, or virtual-virtual, fab, elements in the Fock matrix, since these can force an
iterative solution. For example with CCSD(T), the T3 amplitudes are given by [164]
Dajktk = E P(i/jk)P(a/bc)ta (bcl lei) P(i/jk)P(a/bc)tbi(jkllma)
e m
+P(i/jk)P(a/bc) [ta(bc||jk) + fiat ]
-P~/jk (1abc tbee
-P(i-jk) x (1 im)/imtjkm X Y(1 bae)faetijk (5-19)
m e
The last terms in Equation 5-19 contain fiem and fae elements coupled to Ta. If the
fime or fe elements are not zero, the T3 amplitudes appears on both sides of the
equation and it must be solved iteratively. However, we can use the invariance of
CC theory to orbital rotations among the occupied and virtual spaces to choose to
make a semicanonical transformation of the resultant Fock matrix to simplify the
solution of Equation 5-19. With semicanonical orbitals, the fij (ifj) and fab (afb)
elements are zero, however, the Fock matrix does contain fia terms. Since the fi,




elements are coupled to T2 amplitudes instead of T3's in Equation 5-19, they do
not force an iterative solution.
One close approximation to natural orbitals that avoids the complications of fij
(i:Aj), fab (a~hb), and even fi,, terms in energy calculations is frozen natural orbitals
(FNOs) [165]. In this approach, only the virtual-virtual block of the one particle
density matrix is diagonalized to generate natural orbitals constrained to the virtual
space. Within the full virtual space, FNOs are just a unitary transformation of the
original (e.g., Hartree-Fock) orbitals, so the energy is invariant. However, FNOs can
achieve nearly all of the energy in a smaller space which provides considerable
savings in the resulting calculation. Since some virtual orbitals are excluded from
the resulting calculation or dropped, the energy is no longer invariant. Furthermore,
the active block of the virtual-virtual Fock matrix can be diagonalized, so there are
no fab (a-hb) terms which need to included in the energy calculation. However, the
Fock matrix is not block diagonal with respect to the active and inactive portions,
so there are fab (a54b) terms within the inactive space which are pertinent in
gradient calculations. Some numerical results of FNOs taken from a MBPT(2) wave
function are shown in the next section.
5.5 Illustrative Examples
Table 5-1. Percent of total CCSD and CCSD(T) correlation energies recovered for
different orbital choices. The molecular system is CO with r=
1.12832 A [5].
Percent of virtual space (number of virtual orbitals)
CCSD CCSD(T)
80%(82) 60%(62) 407o(41) _8-0-V 60% 40%
SCF 89.4 77.0 46.7 89.2 76.4 46.6
V. -1 90.0 78.9 53.6
I? 95.7 90.1 48.8
LDA/VWN 89.4 77.1 46.8
PW91 89.4 77.1 46.7
FNO 100.0 99.4 97.1 100.0 99.4 96.9




100,
,90
80 ..... ........." --FN O
" 0- SCF
70......... LDA/VWN
... PW 91
+ Vn-1
+060 ----Vn-a
40,
40 50 60 70 80
percent of virtual space
Figure 5-1. Percent of total CCSD correlation energies recovered for different
orbital choices. The molecular system is CO.
In this section, a number of examples are considered to show the numerical
performance of different orbital choices in a reduced space. In Table 5-1, the
percentage of the correlation energy is tabulated for different choices of the orbital
space and truncations of that space. The molecule is CO described in an
aug-cc-pVTZ basis with all core electrons frozen. The results are also presented
graphically in Figure 5-1. The SCF, LDA/VWN, and PW91 virtual orbitals recover
nearly identical amounts of the CCSD correlation energy for the three truncations
of the space. The 1_ and ln results recover slightly more of correlation energy
in the reduced spaces. However, the FNOs perform much better for both the CCSD
and CCSD(T) correlation methods. For CCSD, FNOs achieve 100.0%, 99.4%, and
97.1% of the correlation energy in 80%, 60%, and 40% of the virtual space,
respectively.




Table 5-2. Percent of total CCSD and CCSD(T) correlation energies recovered for
different orbital choices. The molecular system is C2H4 with rcc=
1.3342 A, rCH = 1.0812 A and aCCH = 121.3 [5].
Percent of virtual space (number of virtual orbitals)
CCSD CCSD(T)
80%(162) 60%(121) 40%(81) 80% 60% 40%
SCF 95.6 78.5 55.6 95.5 77.8 54.6
V I95.8 79.9 61.6
I- 97.3 90.1 48.6
LDA/VWN 95.5 77.0 56.0
PW91 95.5 76.9 56.0
FNO 100.0 99.8 98.6 100.0 99.8 98.5
Table 5-3. Percent of total CCSD and CCSD(T) correlation energies recovered for
different orbital choices. The molecular system is F2 with r -
1.41193 A [5].
Percent of virtual space (number of virtual orbitals)
CCSD CCSD(T)
80%(81) 60%(61) 40%(40) 80% 60% 40%
SCIF 89.1 74.6 52.8 88.9 74.2 52.5
7- 89.6 76.1 52.8
V 96.8 90.0 53.2
LDA/VWN 89.1 74.6 52.8
PW91 89.0 74.5 52.8
FNO 99.8 98.7 93.2 99.8 98.6 92.8
Tables 5-2 through 5-5 present additional results for the C21-4, F2, H20, and
NH3 molecular systems. One additional trend is that the V,,-_ potential performs
better for 80% and 60% reductions of the virtual space, but the V7,_1 potential
performs slightly better for the 40% reduction of the virtual space.
Because total energies are seldom important, it is more constructive to examine
computed energy differences. In Table 5-6, activation energies for the unimolecular
dissociation of the pentazole anion, N5-, are presented. Again the SCF, LDA/VWN,
and PW91 results are quite similar. The results for the FNOs are remarkably good.
For the CCSD(T) method, FNOs with only 40% of the virtual space predict an




Table 5-4. Percent of total CCSD and CCSD(T) correlation energies recovered for
different orbital choices. The molecular system is H20 with rOH ~
0.9572 A and aHOH = 104.52 [5].
Percent of virtual space (number of virtual orbitals)
CCSD CCSD(T)
80%(80) 60%(60) 40%(40) 80% 60% 40%
SCF 85.5 73.8 38.0 85.2 73.2 38.1
-1 87.0 77.1 48.0
-a 95.8 90.6 4.7
LDA/VWN 85.5 73.8 38.0
PW91 85.5 73.7 38.2
FNO 100.0 99.7 98.3 100.0 99.7 98.2
Table 5-5. Percent of total CCSD and CCSD(T) correlation energies recovered for
different orbital choices. The molecular system is NH3 with rNH =
1.0116 A and aNHN = 106.7 [5].
Percent of virtual space (number of virtual orbitals)
CCSD CCSD(T)
80%(100) 60%(75) 40%(50) 80% 60% 40%
SCF 90.9 77.9 47.1 90.6 77.1 46.4
-1 93.0 80.5 59.0
-a 97.4 92.2 17.1
LDA/VWN 90.9 77.8 47.2
PW91 90.9 77.8 47.0
FNO 100.0 99.8 98.7 100.0 99.8 98.6
activation energy of 29.3 kcal mol-1. This is within 1 kcal mol-1 of the CCSD(T)
result for the full virtual space, 28.8 kcal mol-1.




Table 5-6. Activation energies in kcal mo1-' for different orbital choices. The percent of the total activation energy is shown
in parentheses. The molecular system is N 5 at the CCSD(T)/aug-cc-pVTZ ground and transition state optimized
geometries.
Percent of virtual space (number of virtual orbitals)
CCSD CCSD(T)
100%(162) 80%(136) 60%(109) 40%(83) 100 80 60 40
SCF 34.7(100.0) 36.0(103.6) 32.5(93.6) ,,29.2(84.1) 28.8(100.0) 30.4(105.6) 27.0(93.8) 24.3(84.3)
Vn135.8(103.0) 32.2(92.7) 29.2(83.9)
Vn 33.3(95.9) 29.4(84.6) 20.4(58.8)
LDA/VWN 36.0(103.7) 32.6(93.7) 29.3(84.5)
PW91 36.0(103.7) 32.6(93.8) 29.4(84.6)
FNO 34.6(99.6) 35.2(101.3) 35.1(101.2) 28.7(99.6) 29.6(102.6) 29.3(101.6)




CHAPTER 6
CONCLUSIONS
Theoretical chemistry methods have a great potential to study, predict, and
discover high-energy density materials. Often an initial theoretical survey can
identify several promising highly energetic molecules that may be investigated
further. In the more detailed work, theory can closely assist the experimental
efforts. In fact CC methods have produced accurate structures and energetics,
however, one of the bottlenecks in their conventional formulation and application is
the dimension of the molecular orbital basis.
This dissertation has explored several different choices for the orbital space and
the numerical results show that CC methods are largely invariant to different
choices for the virtual orbital space including DFT, and 'V,-, potentials.
However, approximate frozen natural orbitals define a better virtual space of
reduced dimension where much more of the correlation energy is recovered. The
FNO procedure, detailed in Appendix A, constructs a reduced space with natural
orbital information from the full space. Hence, FNOs achieve a large savings in the
time needed per iteration of the CC equations. Although not investigated in this
dissertation, FNOs might be useful in coupled-cluster equation-of-motion
calculations for electronically excited, ionized and attached states. Additional
future work could extend analytical gradient techniques for references with FNO
reductions of the virtual space.




APPENDIX A
COMPUTATIONAL IMPLEMENTATION
The ability to generate frozen natural orbitals has been implemented into the
ACES II suite of computer programs [68]. The following is a summary of the
procedures performed by the xfno ACES member executable. It is important to
note that in the limit of the full virtual space (i.e., 6=0), this procedure results in
canonical Hartree-Fock virtual orbitals.
* The virtual-virtual block of the relaxed one-particle density matrix is
calculated. For MBPT(2), this is:

1 (ijf ac)(ij) bc)
2 (E + Ej fa Ec) (Ei + j Eb Ec)

(A-1)

* Frozen natural orbitals, U, and their occupation numbers are obtained by
diagonalizing this matrix:

UtDabU = u
* The natural orbitals are transformed to the basis of symmetry-adapted
orbitals.

(A-2)

vrt vrt vrt
so C vrt U = so U' (A-3)
* The virtual-virtual block of the Fock matrix is built in the basis of natural
orbitals.
so so vrt vrt
vrt U't so F so U = yrt F' (A-4)
* The active block of the Fock matrix is diagonalized to produce new orbital
energies for the reduced space. The orbital energies for the reduced space are
always larger than the original orbital energies.
vrt 6 vrt 6 vrt 6
vrt 6 Zt vrt 6 F' vrt 6 Z = E' (A-5)
* The orbitals are updated to diagonalize the active virtual block of the Fock
matrix.
vrt 6 vrt 6 vrt 6
so U' vrt 6 Z = so U" (A-6)




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BIOGRAPHICAL SKETCH
Kenneth John Wilson Jr. was born on September 13, 1975 in Youngstown,
Ohio. The city of Youngstown in situated in northeastern Ohio, 120 miles south of
Lake Erie, where much of the nation's steel was produced. Kenneth grew up in
Youngstown, and attended public schools. He graduated fourth in a class of 195
from Woodrow Wilson High School on June 10, 1993. Then he studied at
Youngstown State University, majored in chemistry, and graduated with a Bachelor
of Science, Summa Cum Laude on June 21, 1997.
Kenneth enjoyed his time at the University of Florida and actively participated
in many organizations. These include Mayors' Council (which advises UF's Division
of Housing), Golden Key National Honor Society, Student Senate (which
administers the Student Activity and Service Fee), The Honor Society of Phi Kappa
Phi, and The International Honorary for Leaders in University Apartment
Communities. Off campus, Kenneth served on the Bicycle and Pedestrian Advisory
Committee (which advises the City of Gainesville, Alachua County, and the
Metropolitan Transportation Planning Organization on transportation and
long-range strategic-planning issues). During his service, he was involved with
making the UF campus more bike- and pedestrian-friendly, and authored Student
Senate Bill 2001-1069: Resolution to Improve Pedestrian Safety Along Village
Drive. As one of twenty students during 2001, he was inducted into UF's Hall of
Fame in recognition of outstanding service and achievement in the UF community.
The Hall of Fame is located on the third floor of the Reitz Union and displays
student portraits beginning from 1920.




I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Docto )Of Philosophy. ./
Rodny [ate, Chi
Graduate Research Professor of
Chemistry
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctoro Pilsophy.
N. Yng# Ohrn
Professor of Chemistry
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of o -'o h losophy.
Hfendrik J~ lnkhorst
Professor '6 Physics
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Associate Professor of Chemistry
I certify that I have read this study and that in my opinion it conforms to
acceptable standards of scholarly presentation and is fully adequate, in scope and
quality, as a dissertation for the degree of Doctor of Philosophy.
Associate Professor of Chemistry




This dissertation was submitted to the Graduate Faculty of the Department of
Chemistry in the College of Liberal Arts and Science and to the Graduate School
and was accepted as partial fulfillment of the requirements for the degree of Doctor
of Philosophy.
May 2002
Dean, Graduate School




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