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Potential energy surfaces: the key to structure, dynamics, and thermodynamics. K. D. Jordan. Department of Chemistry. University of Pittsburgh Pittsburgh, PA. ACS PRF Summer School on Computation, Simulation, and Theory in Chemistry,
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Potential energy surfaces: the key to structure, dynamics, and thermodynamics
K. D. Jordan
Department of Chemistry
University of Pittsburgh
Pittsburgh, PA
ACS PRF Summer School on
Computation, Simulation, and Theory in Chemistry,
Chemical Biology, and Materials Chemistry, June 1518, 2005
Potential energy surfaces (PES)
Example – LennardJones (LJ) clusters
21/6σ
E
Two atoms:
R
R
R
ε
repulsion
dispersion (van der Waals)
Multiple atoms  assume pairwise additive:
a
1
2
3
b
2
1
3
Stationary points for all coordinates Xi
Potential energy surface for a twodimensional system, i.e., E(x,y) [from Wales]
Contour map of PES; M = minimum, TS =1st order saddle point, S = 2nd order saddle point
E
E(kJ/mol)
Figures from Energy Landscapes, by D. Wales.
E(kJ/mol)
Easy to find global minimum
Hard to find global minimum
Entropy
unfolded
partially folded
folded
Even though my examples are drawn from cluster systems, the issues considered are relevant for a wide range of other chemical and biological systems, e.g., to the “protein folding” problem. The above figure is from Brooks et al., Science (2001).
Locating transition states and reaction pathways
EF method
Energy (kJ/mol)
Icosahedral
FCC
Icosahedral
Disconnectivity diagram Ar38
(from D. Wales)
Disconnectivity diagram Ar13
(from D. Wales)
Potential energy vs. T, LJ38
solid liquid
FCC Icosahedral
C
C
C vs. T, (H2O)8 (Tharrington and Jordan)
C vs. T, LJ38 (Liu and Jordan)
21
60
6
60
Mass spectrum of (H2O)nH+: magic # at n = 21(from Castleman + Bowen)
Mass spectrum of Cn+: magic # at n = 60 (from Kroto)
Pot. Energy distribution for (H2O)8, T ≈ Tmax
Densities of local minima of (H2O)n clusters
Mass spectra alone tell us very little about the structures.
IR spectra of (H2O)nH+, n = 211, from Duncan, et al., Science, in press
One of the biggest challenges in theoretical/computational chemistry is choosing the suitable approach
Approach to be adopted dictated by the nature of the problem being studied
This will be illustrated by considering the protonated water clusters
secondary
Approaches for modeling
Choice of theoretical approaches for our studies of H+(H2O)n
Quantum Chemistry (electronic structure methods)
Structures responsible for observed spectra
H+(H2O)4
H+(H2O)2
H+(H2O)3
H+(H2O)5
H+(H2O)6
H+(H2O)8
For the n = 5  8 clusters, these are not the global minimum isomers.
Accounting for finite temperature on cluster stability
Optimize geometries
Eel (T=0)
Eel(T=0)+ ZPE
E(T = T’)
H(T=T’)
G(T=T’)
From electronic structure calculations
Account for vibrational zeropoint energy
Calculate harmonic frequencies
Population of excited vibrational,
rotational levels
Account for PΔV = ΔnRT (ideal gas)
Include entropy
3.
2.
5.
6.
4
3
4
1, 2
5
Eele
Eele+ZPE
G(50K)
G(100K)
G(150K)
G(200K)
6
(H2O)6H+
Isomers with dangling water molecules (low frequencies) favored by ZPE and by entropy
Zundeltype ion dominates under the experimental conditions, T 150 K.
Comparison of calculated and measured vibrational spectra of H+(H2O)6
Theory
Intensity
Expt.
Intensity
vibrational spectra of H+(H2O)n, n = 627
Collapse to a single line in the free OH stretch region
freeOH region of spectra reflect structural transitions
at n = 12 and n = 21(Shin et al., Science, 2004)
Lowestenergy n=21 structure found in ab initio geometry optimizations
Dodecahedron with H3O+ on surface (blue) and H2O (purple) inside cage
4 Hbonds with interior H2O
causes a rearrangment of the Hbonding in the dodecahedron
there are only 9 freeOH
groups (Castleman\'s experiments suggested 10)
all freeOH associated with AAD waters  explains single lines in free OH stretch
If the excess proton placed on interior water, it rapidly jumps to surface.
Mass spec.
Mass spec.
source
Predissociation spectroscopy
H+(H2O)n H+(H2O)n1 + H2O
Calculated vs. expt. spectra of magic # cluster.
No transitions observed in H3O+ OH stretch region
If the ion does not fall apart on the timescale of the experiment, no signal will be observed.
210
free OH
Eigen OH
106 s.
190
Tm
T(K)
170
102 s.
150
τ
130
without Ar
with Ar
These problems illustrate the interplay between structure, spectra, and dynamics inherent in much of today’s research
Several transitions of the H+(H2O)n clusters are not well described in the harmonic approximation
x=(RRe)/Re
ωe = harmonic frequency
ωexe, ωeye = first two anharmonicity constants
Be = rotational constant
αe = vibr.rot. coupling
ωe = sqrt(4ao*Be)
αe = (a1 + 1)(6Be2/ ωe)
ωexe = (5a12/4 – a2)(3Be/2)
Depends on 3rd and 4th derivatives
Dunham expansion: unique mapping between 1D potential and the spectroscopic parameters
This mapping is lost for polyatomic molecules
Approaches for treating anharmonicity
CH3NO2(H2O) – an example of important offdiagonal vibrational anharmonicity
expt.
theoryharmonic
OH stretch
CH stretch
theory  anharmonic
From Johnson, Sibert, Jordan and Myshakin, 2004
(H2O)2 – an example illustrating the importance of vibrational anharmonicity of frequencies, ZPE, geometry
donor
acceptor
donor
Frequencies calculated using the MP2 method.
Anharmonicities calculated using 2nd order vibrational PT.
Excellent agreement between the calculated anh. frequencies and experiment.
Intermolecular vibrations
Changes in bond lengths of (H2O)2 upon vibrationally averaging
E
Re
Ro
R
Challenges facing electronic structure theory
Some considerations concerning model potentials
Many of the issues can be illustrated by considerations of models for water.
+q
M, 2q
+q
Water dimer: interaction energies (kcal/mol)
O
H
MP2 – inplane
Inplane electrostatic potential of the water monomer from MP2 ab initio calculations from and from the DC water model. Distances in Å.
Outer contour = 0.005 au = 3 kcal/mol
0.005
0.005
DC model – in plane
M
0.005
DC model: q = +0.519 H atoms, 1.038 M site, 0.215 Å from the O atom.
0.005
Inplane electrostatic potential: DC – MP2. Outer blue contour 0.0005 au = 0.3 kcal/mol. Distances in Å.
Perp.toplane electrostatic potential: DC – MP2. Outer black contour 0.0005 au = 0.3 kcal/mol. Distances in Å.
In these figures the part of the electrostatic potential near the atoms has been cut out.
A threepoint charge model cannot realistically describe the electrostatic potential potential of water!!
Yet, nearly all simulations of water, ice, and biomolecules in water use models with simple point charge representations of the charge distribution.
Inplane
Differences between the electrostatic potentials from a distributed multipole analysis with moments through the quadrupole on each atom and from MP2 level calculations.
Overall the agreement is excellent except for short distances.
0
Perp. to plane
0
Inplane electrostatic potential: Amoeba – MP2. Outer blue contour 0.0005 au = 0.3 kcal/mol. Distances in Å.
0
Perp.toplane electrostatic potential: Amoeba – MP2. Outer light blue contour 0.0005 au = 0.3 kcal/mol. Distances in Å.
0
Amoeba should give results identical to GDMA. Differences due to change in HOH angle and scaling of the atomic quadrupoles.
More on polarization interactions
+
+
A
μAB


+
+
μBA
B


Inert gas clusters – manybody effects dominated by dispersion
Water clusters – manybody effects dominated by polarization
+
C

μij – dipole induced on i by charges on j
Isolated water monomer – dipole moment = 1.85 D
Water molecule in liquid water – dipole moment ~ 2.6 D
μAB in turninduces a dipole moment on B. Infinite series!
Effective 2body potentials for water, e.g. TIP4P and SPC/E, have charges that give a dipole significantly larger than experiment for the monomer
The orbital picture reconsidered.
Issues connected with unfilled orbitals
Potential energy curves of CH3Cl and CH3Cl
Vibrational excitation cross sections for two vibrations of CH3Cl.
The peaks are due to resonances (temporary anion states).
From P. Burrow.
Electrons bound in electrostatic potentials
The electron is so extended, that it should be possible to develop a oneelectron model approach
An excess electron bound to a (H2O)6 chain
Cannot simply add a C/R6 term, due to extended nature of excess electron.
We have developed a Drude model of excesselectron molecule interactions.
+q q charges +q, q coupled through a force constant k
R The position of the q charge is kept fixed.
In the presence of a field, the system has a polarizability of q2/k.
An electron couples to the Drude oscillator via qr∙R/r3
,
O
H
Drude model based on the DangChang water model
H charge = 0.519e
M site: 0.215 Å from O atom.
Negative charge (1.038e) plus Drude oscillator with q2/k = α = 1.444 Å3
Damping coefficient scaled so that model potential CI energy reproduces ab initio CCSD(T) result for (H2O)2
b
r  position of electron
R  displacement of the Drude oscillator
Wavefunction:
Electron orbitals described in
terms of s, p Gaussians. { }
in “MO” basis set
3D harmonic oscillator functions { }
Multiple Drude Oscillators:
Basis set:
Several strategies for solving
Surface state and interior electron bound states of (H2O)20
Considerable interest in these speciesin light of recent work from the Neumark and Zewail groups.
The anion is not bound in the KT and HartreeFock approximations. Electron binding is a result of correlation effects which cause a large contraction of the excess electron
Geometries provided by M. Head_Gordon.