Computational study on NiAl: ground state, structure, and spectroscopic constants using densityfunctional theory
T. Cundari1, S.S. Janardan1
1University
of North Texas, Department of Chemistry
Abstract
The least computational expensive Hamiltonian (methods) and wavefunction (basis sets) for diatomic nickel-aluminum (NiAl), was calculated,
along with the ground state, bond length, frequency, and binding energy. Researchers believe nickel-aluminum can be used in high performance
turbine jet engines, in place of aluminum-coated nickel. The effects of basis functions under increasing levels of theory were compared to methods
under increasing levels of theory. The free energy at two possible ground states, doublet or quartet, of nickel-aluminum was compared. The results
consistently show the ground-state of nickel-aluminum is the doublet, and therefore has a bond between two valence electrons, one from each nickel
and aluminum, and one unpaired electron. The data converged when using large number of basis sets, but did not produce precise results when
compared with different methods. Results suggest the density-functional theory (DFT) and an augmented correlation consistent basis set are needed,
at minimum, to properly optimize nickel-aluminum. The least computationally expensive, most precise basis set/method combination for diatomic
NiAl allows for further research in the least computationally expensive combination, of method and basis set, for large microclusters of NiAl, which
could be used in engines as well.
Keywords: NiAl; level of theory; Hamiltonian; wavefunction; ground-state
I.
Introduction
Nickel-based superalloys are widely used in high-performance
jet engines, and thus constitute an important area of research in
material science, among both the experimental and modeling
communities. These materials display extreme durability at high
temperatures and possess the ability to precisely control their
mechanical properties via the addition of many alloying metals. The
jet engine application requires that alloys be both heat and corrosion
resistant, and able to withstand creep and friction [1]. Nickel has
traditionally been used in aerospace alloys [2], but it has a low
oxidation resistance and requires a thermal barrier coating (TBC) [3].
Recent studies have combined nickel and aluminum (along with other
minor components) using high-temperature synthesis [1].
The alloy NiAl has a heat resistance of 1640 ºC - 1677 ºC [1, 35] before the alloy starts to melt, an increase over traditional Ni
alloys [1, 2]. Nickel-aluminum alloys display extremely high
resistance to corrosion and oxidation compared to nickel [1, 3-5].
Recent calculations on molecular NiAl materials include a report by
Oymak and Erkoc [6] using density functional theory (DFT) and
effective core potential (ECP) methods. The geometries for AlkTilNim
clusters were calculated for (k + l + m = 2, 3), examining geometries,
vibrational spectra, dissociation, and electronic states. The binding
energy of the diatomic NiAl, equilibrium interatomic separation, and
stretching frequency were calculated to be 3.35 eV, 2.533 Å, and
263.5 cm-1, respectively [6]. NiAl was found to have greater binding
energy as compared to diatomic AlTi and diatomic NiTi, and a
smaller bond length and stretching frequency as compared to these
same molecules. When NiAl was compared to all AlkTilNim clusters
studied by Oymak and Erkoc, NiAl was found to have the largest
highest occupied molecular orbital and lowest unoccupied molecular
orbital (HUMO-LUMO) gap, typically used as an indicator of kinetic
stability, as compared to AlTi and NiAl.
More recent calculations by Fengyou et al. [7] found the bond
length between nickel and aluminum in NiAl to be 2.19 Å and the
binding energy to be 1.23 eV. Fengyou et al. used the B3PW91
density functional, as did Oymak and Erkoc [6], but Fengyou et al.
used a relatively small LANL2DZ valence basis set, possibly
accounting for the large difference in numbers between the two
calculations. Out of all NiAl microclusters (combinations of 5, 13,
and 52 nickel and aluminum molecules) analyzed, NiAl was found by
Fengyou et al. to have the shortest bond length between the nickel
and aluminum. The frequency of diatomic NiAl was calculated to be
374 cm-1, a large difference versus the calculations of Oymak et al.
However, the frequency of NiAl was the highest among all nickelaluminum clusters tested by these researchers [7].
In a report published at approximately the same time, Belcher et
al. [8] found the bond length of diatomic NiAl to be 2.20 Å, quite
similar to the results of Fengyou et al. [7]. Belcher et al. [8] used
plane wave DFTs and pseudopotential methods within the Vienna
Ab-initio Simulation Package (VASP) code; AlX (X = transition
metal) diatomics were compared, and NiAl was found to have the
shortest bond length as compared with aluminum-gold (AlAu) and
aluminum-copper (AlCu), but NiAl still had a smaller HUMOLUMO gap than AlAu and AlCu. However, the calculation for the
binding energy of NiAl came to be -3.30 eV, showing a large gap in
data when compared to other experiments [8].
The most recent calculations by Ouyang et al. [9] did use DFT,
more specifically a hybrid density functional method B3LYP. The
preferred basis set used here was the 6-311+G(2d) and the Lan12DZ
basis set. The calculated bond length is similar to the calculation done
by Oymak et al. [6] at 2.39 Å, but the basis set used is the same as
Fengyou et al. [7]. The frequency was calculated to be 294 cm-1,
similar to Oymak et al. [6] by have a large difference in numbers
comparatively. Calculation of the binding energy yielded 2.24 eV,
half way between previous calculations of Oymak and Erkoc [6] and
Fengyou et al. [7]. The ground state of NiAl was also examined, and
compared o aluminum-platinum (AlPt) and aluminum-palladium
(AlPd). The ground state of NiAl was found to be very similar to
AlPt and AlPd, although few specifics were discussed [9].
The data from previous research indicate that the
wavefunctions/basis set used in NiAl calculations influences the
results greatly. The effect of different basis set has yet to be carefully
assessed for NiAl materials to produce the “best” basis set with
respect to computational accuracy and cost. This is quite surprising in
terms of the great interest in Ni-Al intermetallics, particularly given
the popularity of utilizing first-principles calculations on small cluster
to derive interatomic potentials that may be used to perform largerscale atomistic simulations. This paper is divided into three main
parts. The first part discusses the different Hamiltonians/method (Ĥ),
wavefunctions/basis sets (ψ), and computing methods used to
III. Results and Discussion
a. Ground State of NiAl
Surprisingly, one issue that has not been fully addressed in the prior
literature is the ground state of NiAl. Nickel has a triplet (3F) ground
state and aluminum is a doublet (2P), so that the two most plausible
224/aug-cc-pVQZ
163/cc-pVQZ
143/aug-cc-pVTZ
102/cc-pVTZ
60/VTZ
47/TZV
40/3-21G
42/pVDZ
35
30
25
20
15
10
5
0
34/VDZ
II. Computational Methods
Calculations employed the Gaussian 03 package. For each basis
set and method, four calculations were performed: Ni (triplet ground
state), Al (doublet ground state), NiAl doublet, and NiAl quartet. The
main methodology used in the present calculations was density
functional theory (DFT). However, wavefunction-based 2nd order
Møller–Plesset perturbation theory (MP2), Hartree–Fock (HF), and
coupled cluster (CC) techniques were also used to compare to and
contrast with DFT.
Multiple basis sets and Hamiltonians were used in assessing the
“best” level of theory, looking for convergence of properties for
diatomic NiAl. The HF method provides a starting point for the postHartree-Fock methods, viz CC and MP2 methods. Two CC methods
were evaluated for the calculations on NiAl: coupled cluster with
single and double excitations (CCSD) and coupled cluster with
single, double and perturbative triple excitations (CCSD(T)). The
latter proved to be too expensive using the extended basis sets needed
to describe NiAl and hence this work focuses on the CCSD
methodology. Two DFT methods were extensively evaluated with
different basis sets used for the calculations of NiAl: the Becke,
three-parameter, Lee-Yang-Parr exchange-correlation functional
(B3LYP) and B971, the latter being the most popular in molecular
calculations on transition metal compounds and the former being
evaluated as superior with respect to thermodynamic predictions in a
previous survey of 3d transition metal complexes and 4? functionals.
As with increasing sophistication in the Hamiltonians, the basis
sets may be enhanced in complexity until convergence in properties
is seen. The quality of atom-centered basis sets for molecular
calculations are, in general, directly correlated with the number of
contracted Gaussian-type orbitals (cGTOS). For example, the 3-21G
split-valence basis set is one of the smallest basis set evaluated here,
comprising only 40 cGTOS for NiAl. The VDZ[10] and VTZ[10]
basis sets possess 34 and 60 cGTOS, respectively. To add more
flexibility, a polarized double-zeta basis set (pVDZ)[10], 42 cGTOS
was evaluated. The last of the computational inexpensive basis sets
utilized in this research is the TZV[11], which is a split-valence basis
set and has 47 cGTOS for NiAl.
One issue that has traditionally plagued transition metal
computational chemistry is the absence of basis sets that have been
methodically improved to allow one to more smoothly approach the
basis set limits for the calculation of various molecular properties.
Correlation consistent (cc) basis sets were originally developed by
Dunning and coworkers for main group elements and recently
extended to transition metals by Peterson et al. The present
calculations used cc-pVTZ, cc-pVQZ, cc-pV5Z basis sets, with 102,
163, and 248 cGTOS, respectively, for NiAl. Augmenting these
further by the addition of diffuse functions to the correlation
consistent basis sets creates the largest basis sets used in the present
calculations: aug-cc-pVTZ, aug-cc-pVQZ, aug-cc-pV5Z), with 143,
224, and 333 cGTOS, respectively for the diatomic.
Diatomic NiAl was geometry optimized, and its equilibrium
molecular properties calculated at the aforementioned levels of theory
and using both (B3LYP and B971) DFT methods. The NiAl diatomic
was studied both in the quartet and doublet spin states for each level
of theory, along with nickel (triplet) and aluminum (doublet)
separately.
ground state multiplicities for NiAl arises from high-spin (4NiAl) and
low-spin (2NiAl) coupling of the constituent atoms. We note that in
these calculations that no symmetry restrictions were placed on the
wavefunction and all simulations were spin-polarized (unrestricted).
No evidence of significant spin contamination was observed in the
calculations.
Doublet-Quartet Gap (Kcal/mol)
calculate the properties of NiAl. The second part analyzes the results
of the calculations in relation to the difference in the lowest energy
doublet and quartet states (GDQ), equilibrium bond lengths (re),
equilibrium stretching frequency (e), and binding energy (De) for
different levels of theory. The last part contains the conclusions.
Number of Basis Functions
Figure 1. Doublet vs. quartet free energy difference (a.u.) in diatomic
NiAl. The number of contracted Gaussians is denoted along with the
basis set name. The B3LYP functional is used for these simulations.
The calculation of the relative energy of the most stable doublet and
quartet spin states states of NiAl produced consistent results for
nearly all levels of theory, apart from the small 3-21G basis sets, as
seen in Figure 1. Doublet NiAl consistently yielded a lower free
energy than quartet NiAl. When GDQ is analyzed for the B3LYP
method and all basis sets, convergence to a free energy difference of
ca. 25½ kcal/mol (~1.1 eV) is seen, with the doublet state always
being lower in free energy. The aug-cc-pVQZ basis set had a value of
25.6187 kcal/mol for the doublet quartet split. The DFT calculations
thus point to the most probable spin state of NiAl as a doublet and
this suggests a significant degree of Ni-Al covalent bonding via spin
pairing, Scheme 1.
Ni
Ni
(3F)
+
4
NiAl
Al
Al
(2P)
Ni
Al
2
NiAl
Scheme 1
b. Equilibrium Bond Length
Structural data remains one of the most fundamental parameters
that is either incorporated into empirical potentials or which is
required to be faithfully reproduced by said potentials. The NiAl
equilibrium bond length (re) was calculated for each basis set and
both the B3LYP and B97-1 functionals. Similar to the convergence
patterns for basis sets and methods, re was seen to converge at higher
theory level basis sets for both DFT methods, Figure 2. However,
while re convergence required a large basis set, Figure 2, it did not
require the largest basis sets evaluated. For consistency, however, the
calculated re with the largest basis set - aug-cc-pVQZ – and the
B3LYP functional = 2.3744Å. This result is comparable with other
224/AUG-CC-pVQZ
163/CC-pVQZ
143/AUG-CC-pVTZ
102/CC-pVTZ
60/VTZ
47/TZV
42/pVDZ
40/3-21g
500
450
400
350
300
250
200
150
100
50
0
34/VDZ
Fundamental Frequency (cm-1)
Fig. 2 Bond length (Å) versus Basis Functions in B3LYP
Number of Basis Functions
Fig. 3 Fundamental Frequency versus Basis Functions in B3LYP
c. Equilibrium Stretching Frequency
The energetic costs of distorting atoms from their equilibrium
positions, whether it is represented by the elastic constants or phonon
modes of a solid material or the vibrational stretching frequencies of
a molecular entity, is one of the most important quantities in
developing and utilizing empirical potentials. Such fundamental
quantities necessarily impact and inform material properties such as
bulk, shear and Young’s moduli.
The equilibrium stretching frequency (e), was calculated for all
basis sets and both DFT methods. The basis set convergence patterns
of the frequencies matched all properties discussed previously, Fig. 3.
The calculated e, akin to re, was less basis set dependent than the
free energy. Using the largest aug-cc-pVQZ basis set and B3LYP, the
frequency was calculated to be 283 cm-1. This theoretical value is not
comparable to previous research seen, since the other values
consisted of 263.5 cm-1 [6], 374 cm-1 [7], and 294 cm-1 [9].
d. Binding Energy
224/AUG-CC-pVQZ
163/CC-pVQZ
143/AUG-CC-pVTZ
102/CC-pVTZ
60/VTZ
47/TZV
42/pVDZ
34/VDZ
Binding Energy (Kcal/mol)
224/AUG-CC-pVQZ
163/CC-pVQZ
143/AUG-CC-pVTZ
102/CC-pVTZ
60/VTZ
47/TZV
42/pVDZ
40/3-21g
Number of Basis Functions
40/3-21g
Number of Basis Functions
2.45
2.4
2.35
2.3
2.25
2.2
2.15
2.1
2.05
2
1.95
1.9
34/VDZ
Bond Length (Å)
reports, which suggested re = 2.533 Å[6] and 2.3875 Å [9].
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
Fig. 6 Binding Energy versus Basis Functions in B3LYP
Each method and basis set was calculated four times, once each
with Ni, Al, NiAl-quartet and Ni-doublet. Based on the equation Ni +
Al  NiAl, the binding energy can be calculated from the free
energy results. Due to errors in the computing process very few basis
sets had all four calculations run, and therefore produce the
calculation for binding energy. As with bond energy and frequency,
the convergence pattern remained the same when comparing binding
energy across basis sets and methods, fig. 6. Similar to the bond
length and frequency, the convergence of the binging energy comes
to around 102 basis sets. Although, not enough data was found to
accurately claim this. The final result for binding energy was
calculated using the AUG-cc-pVQZ basis set and DFT method,
producing -44.2 (kcal/mol).
IV. Conclusion
The most efficient basis set was determined to be AUG-CCpVQZ because of the high accuracy to computational cost ratio. The
best method was determined to be density-functional theory, because
the other methods did not have accurate enough results. The ground
state of NiAl was determined to be a doublet, with one electron from
nickel and aluminum forming a magnetic bond, leaving one electron
free. Further exploration of the doublet NiAl revealed the bond
length, frequency, and binding energy to be 2.37Å, 283 cm-1, and 44.2 (kcal/mol) respectively.
The most efficient basis set and method were only determined
for diatomic NiAl. A common basis set between all variations of
NiAl still needs further research. Small microclusters in ratios of 3:1
nickel to aluminum can be built up in multiples and evaluated to find
a common basis set and method. The common basis set for diatomic
NiAl can also be used when researching further into the properties of
NiAl (e.g. ductile-to-brittle transition temperatures).
V.
Acknowledgements
The authors thank the Center for Advanced Scientific
Computing and Modeling (CASCaM) for providing computing
resources (NSF grant CHE-??????), and Dr. Nathan J. DeYonker,
formerly of the UNT Chemistry Department, for his assistance. The
authors acknowledge financial support from the Air Force Research
Laboratory (Contract No. FA8650-08-C-5226). The authors also
thank the Texas Academy of Mathematics and Science for
opportunities and scholarships.
VI. References
[1] Pascal, C., Marin-Ayral, R. M., Tédenac, J.C., & Merlet, C.
(2003). Combustion synthesis: a new route for repair of gas
turbine components—principles and metallurgical structure
in the NiAl/RBD61/superalloy junction. Materials Science
and Engineering A, 341, 144-151.
[2] Johnson, B. J., Kennedy, F. E., & Baker, I. (1996). Dry sliding
wear of NiAl. Wear. 192, 241-247.
[3] Guo, H., Sun, L., Li, H., & Gong, S. (2008). High temperature
oxidation behavior of hafnium modified NiAl bond coat in
EB-PVD thermal barrier coating system. Thin Solid Films,
516, 5732–5735.
[4] Darolia, R. (2000).Ductility and fracture toughness issues related
to implementation of NiAl for gas turbine applications.
Intermetallics. 8, 1321-1327.
[5] Pomeroy, M. J. (2005).Coatings for gas turbine materials and long
term stability issues. Materials and Design. 26, 223-231.
[6] Oymak, H., & Erkoc, S (2002). Structural and electronic
properties of Alk Til Nim microclusters: Densityfunctional-theory calculations. Physical Review A. 66,
033202.
[7] Fengyou, H., Yongfang, Z., Xinying,, L., & Fengli, L. (2007). A
density-functional study of nickel/aluminum microclusters.
Journal of Molecular Structure: Theochem, 807, 153–158.
[8] Belcher, D. R., Radny, M. W., & King, B. V. (2007). Structure
and stability of small bimetallic Al-based clusters: An ab
initio DFT study. School of Mathematical and Physical
Sciences, The University of Newcastle, Callaghan,
Australia. Materials Transactions, 48(4), 689-692.
[0] Rey, C., Garcıa-Rodeja, J., & Gallego, L. J. (1996). Computer
simulation of the ground-state atomic configurations of Ni-
Al clusters using the embedded-atom model. Physical
Review B. 54, 2942-2948.
[9] Ouyang, Y., Wang, J., Hou, Y., Zhong, X., Du, Y., & Feng, Y.
(2008). First principle study of AlX (X=3d, 4d, 5d elements
and Lu) dimer. Journal of chemical physics. 128, 074305.
[10] A. Schafer, H. Horn and R. Ahlrichs, J. Chem. Phys. 97, 2571
(1992).
[11] A. Schafer, C. Huber and R. Ahlrichs,J. Chem. Phys. 100, 5829
(1994).
Bailey, M. S., Wilson, N. T., Wilson, C., & Johnston, R. L. (2003).
Structures, stabilities and ordering in Ni-Al nanoalloy
clusters. Eur. Phys. J. D, 25, 41-55.
Krajci, M., & Hafner, J (2002). Covalent bonding and bandgap
formation in transition-metal aluminides: di-aluminides of
group VIII transition metals. Journal of Physics:
Condensed Matter. 14, 5755–5783.
Reddy, B. V., Deevi, S. C., Lilly, A. C., & Jena, P (2001). Electronic
structure of sub-stoichiometric iron aluminide clusters.
Journal of Physics: Condensed Matter, 13, 8363–8373.
Reddy, B. V., Sastry, D. H., Deevi, S. C., & Khanna, S. N. (2001).
Magnetic coupling and site occupancy of impurities in
Fe3Al. Physical Review B. 64, 224419.
Rexer, E. F., Jellinek, J., Krissinel, E. B., Parks, E. K., & Riley, S. J.
(2002). Theoretical and experimental studies of the
structures of 12-, 13-, and 14-atom bimetallic
nickel/aluminum clusters. Journal of Chemical Physics.
117,
82-94