Introduction to Computational Chemistry organic-chemistry reaction mechanism: What is \wrong" with this? ... Intro Computational Chemistry Prof. Dr. B. Hartke, Kiel University,

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Introduction toComputational Chemistry

(MNF-chem0407)

English translation of German version from 19. Nov. 2013,

with changes/extensions up to May 22, 2018

Prof. Dr. Bernd Hartke

Theoretical Chemistry

Christian-Albrechts-University Kiel

Institute for Physical Chemistry

Max-Eyth-Str. 2

Ground floor, Room 29

Tel.: 0431/880-2753

[email protected]

http://ravel.pctc.uni-kiel.de

office hours: any time by appointment

Intro Computational Chemistry • Prof. Dr. B. Hartke, Kiel University, [email protected]

Contents

• general introduction

• classical-mechanical molecular dynamics (MD), force fields

• electronic structure theory

– basics of quantum mechanics

– effective 1-particle model (Hartree-Fock)

– molecular structures and properties

– Density Functional Theory (DFT)

– explicit electron correlation (CI, CC, MPn)

– excited states

– semiempirical methods

• ab-initio-MD and wavepacket dynamics

• outlook

Aims

• What can be calculated?

• How is it calculated? (models, assumptions)

• Results: how accurate? how reliable?

2


desirable prerequisites

• Mathematics for Chemists 1,2 (MNF-chem0102/0202):

– representations of vectors/operators/funktions in a basis

– matrix eigenvalues and eigenvectors

– (numerical) solution of integrals and differential equations

• Physical Chemistry 2 “Structure of matter” (MNF-chem0304):

– fundamental quantum mechanics (Schrodinger equation, operators)

– quantum mechanics for simple models (particle in a box, rigid rotor,harmonic oszillator, hydrogen atom)

– many-electron atoms, periodic table of elements

– chemical bonding concepts

related, advanced courses

• optional module MNF-chem5014: “Numerical Mathematics for Chemists”numerical treatment of integrals, differential equations, systems of linearequations, matrix eigenvalue problems, etc.

• elective master module MNF-chem1004D:

lecture:

– some mathematical background to electronic structure theory

– solving the time-dependent Schrodinger equation (wavepackets)

practical course:

– brief introduction to programming and numerical mathematics

– developing and using self-written programs for: classical mechanics,solving the time-independent Schrodinger equation by expansion inbasis functions, wavepacket propagation

• elective master module MNF-chem3005D:

– using big program packages

– introduction to computational chemistry research

3


Literature recommendations:

Force-field Molecular Dynamics:

• A. R. Leach: “Molecular Modelling: Principles and Applications”, 2ndedition, Pearson/Prentice-Hall, 2001:survey over everything, w/o going into too many details

• J. M. Haile: “Molecular Dynamics Simulation – Elementary Methods”,Wiley, 1997: basic introduction into the main topics

• D. Frenkel und B. Smit: “Unterstanding Molecular Simulation”, 2ndedition, Academic Press, 2002:advanced introduction including more difficult techniques

Quantum chemistry:

• P. W. Atkins and R. S. Friedman: ”‘Molecular Quantum Mechanics”’,4th edition, Oxford University Press, 2004:standard textbook on quantum mechanics in chemistry; not providingmuch on how molecular electronic structure calculations are actuallydone.

• J. Simons and J. Nichols: ”‘Quantum Mechanics in Chemistry”’, OxfordUniversity Press, New York, 1997:another standard textbook on quantum mechanics in chemistry; moredetails on methods in actual use.

• F. Jensen: “Introduction to Computational Chemistry”, 2nd edition,Wiley, 2007:advanced textbook providing more details relevant for real-life molec-ular calculations; also contains chapters on force fields and classical-mechanical molecular dynamics.

• A. Szabo and N. S. Ostlund: ”‘Modern Quantum Chemistry”’, revised1st edition, McGraw-Hill, 1989:older textbook, but still the best derivation of Hartree-Fock in all thetechnical details.

• W. Koch and M. C. Holthausen: ”‘A Chemist’s Guide to DensityFunctional Theory”’, Wiley, 2001:good and easily accessible textbook for DFT users, also containing moregeneral basics.

4


• T. Heine, J.-O. Joswig and A. Gelessus: “Computational ChemistryWorkbook — Learning Through Examples”, Wiley, 2009:good introduction into several fundamental topics, with explanationsand do-it-yourself calculations (with pencil and paper, or with simpleprovided programs).

• E. G. Lewars: “Computational Chemistry”, Springer, 2011:recommended; good fit to this lecture; comprehensive but detailed; goodfor beginners but still exact; shows several calculations in great detail;many examples, many practical tips.

• J. H. Jensen: “Molecular Modeling Basics”, CRC Press / Taylor&Francis,2010:small booklet (160 p.) but recommended; manages to combine explain-ing the background (including equations) with practical tips for realcalculations (including actual input/output, for GAMESS).

• L. Piela: “Ideas of Quantum Chemistry”, Elsevier, 2007:thick book (1087 pages), spanning everything from basic quantum me-chanics (including its history, relativistics, etc.) via many details andexamples up to presentations of the most relevant quantum chemistrymethods; recommended despite its unconventional style and the occa-sionally bumpy language.

• C. Trindle und D. Shillady: “Electronic structure modeling — connectionbetween theory and software”, CRC Press / Taylor&Francis, 2008:covers everything from basic quantum mechanics to high-end methods likeCCSD(T); illustrates many things with outputs from actual calculations(a simple SCF demonstration program, Gaussian, ADF).

• J. J. W. McDouall: “Computational Quantum Chemistry”, RSC Pub-lishing, 2013:mid-size book (270 p.), but still comprehensive and detailed introduc-tion into all the important topics; clear presentation, in the first partsexplicitly addressing beginners.

• D. B. Cook: “Quantum Chemistry – a unified approach”, ImperialCollege Press, 2008:a theory textbook of a different kind: interesting and partially successfulattempt to connect organic/inorganic concepts (Lewis structures, sp3

hybridization) with actual theory concepts (orbitals, electron densities).

5


Quantum wavepacket dynamics:

• D. J. Tannor: ”‘Introduction to quantum mechanics: a time-dependentperspective”’, University Science Books, Sausalito (CA), 2007:the first textbook on wavepacket dynamics; probably too advanced forabsolute beginners.

• R. Schinke: ”’Photodissociation dynamics”’, Cambridge University Press,Cambridge, 1993 (paperback version of 1995 still obtainable):despite the somewhat misleading title, this book provides a good andaccessible introduction into the fundamentals of quantum dynamics,including the interaction with electromagnetic fields.

• J. Z. H. Zhang: ”’Theory and application of quantum molecular dy-namics”’, World Scientific, Singapore, 1999:good and general but partially difficult and too short presentation, includ-ing interaction with light, scattering theory, semiclassics, rate constantcalculations, etc.

• G. C. Schatz und M. A. Ratner: ”‘Quantum mechanics in chemistry”’,Dover, Mineola, 1993/2002:several chapters on different topics, besides quantum chemistry alsotime-dependent perturbation theory, interaction with light, scatteringtheory, correlation functions, time-dependent simulation of spectra, rateconstant calculations.

6


General status of theoretical chemistry

• basic theory completely known: quantum mechanics(plus special relativity for heavy elements,for motion of nuclei classical mechanics often sufficient);

• validity of this theoretical basis not in doubt anymore;

• practical recipes to calculate everything of chemical interest are known.

⇒ remaining issues in Theoretical Chemistry:practical problems in calculations:

• exact analytical solutions only possible for a few model systems (cf. PC2)⇒ numerical methods necessary;problem: extreme accuracy requirements:

– Quantum chemistry: chemical bond (1–200 kJ/mol) is only a smallperturbation of the interaction between electrons and nuclei (100000-1000000 kJ/mol)⇒ max. error ≤ 1:105, else error � chemistry.

– reaction dynamics: exponential dependence betweeen rate constantsand energy differences

• frequently extremely high computational expense:

– Quantum chemistry: steep increase of computational expense withsystem size n for traditional methods (n3 – en)

– reaction dynamics: vast scale discrepancies (e.g. in time), funda-mental difficulties with everything that involves entropy

– many-particle systems → “curse of dimensionality”(exponential or factorial size scaling)

Nevertheless, computational chemistry today is practically relevant:

Quantum chemistry: routine calculations with “chemical accuracy” for chem-ically relevant systems (many dozens to several hundreds of atoms);linearly scaling methods in development;

Molecular dynamcis: Simulations for small biochemical systems (a fewmillions of atoms) for a few milliseconds (billions of time steps).

7


Scale discrepancies:

Length: typical molecular distances: 1 A = 100 pm = 10−7 mm

1 A

1 mm≈ 1 mm

10 km(1)

With a velocity of 1mms one needs 4 months to cover 10 km.

or: watch the seconds ticking for 1 year. . .

Time: order of magnitude for molecular movements: 1 fs = 10−15 sslow chemical processes (protein folding): milliseconds–seconds

1 fs

1 s≈ 1 s

32 million years(2)

= watch the seconds ticking between now and formation of the Alps.Light travels only 300 nm per 1 fs; it needs 3333 fs for 1 mm.

Particle number: Simulation: single molecules;macroscopical amount: 1 mol ≈ 6 · 1023 molecules

1 molecule

1 mol≈ 1 microsecond

32 billion years(3)

= count the microseconds between now and the Big Bang (13.7 billionyears), two times.

other units:

(rest) mass of one electron: me ≈ 10−30 kg (4)

quantum of action (Planck’s constant): ~ =h

2π≈ 10−34 J s (5)

Consequences:

• Quantum mechanics

• all scale discrepancies = fundamental gaps in calculations

8


Size scaling of computational expense:

(Setting 1s CPU time for 5 particles is an ad-hoc assumption;the scaling, however, is very real)

Scaling → linear cubic exponential

particle number

5 1 s 1 s 1 s10 2 s 8 s 2.5 min20 4 s 1 min 1 months40 8 s 8.5 min 50 mio years80 16 s 1 h 8 min —160 32 s 9 h —320 1 min 3 days —

⇒ size scaling of algorithms is practically very relevant!

• can make the difference between a quick calculation and an impossibleone;

• even seemingly small changes in size scaling can be important.

9


Chemical reactions as viewed by theory

Typical organic-chemistry reaction mechanism:

What is “wrong” with this?

• in electronic structure theory, there is nothing that directly correspondsto lines indicating electron pairs in Lewis structures or to movementsof electrons/electron pairs indicated by curly arrows(however, specially constructed orbitals can look similar, cf. Knizia/Klein,Angew. Chem. 127 (2015) 5609)

• molecular electron density distributions are strongly dominated from theatomic ones

• valence states (sp3, sp2,. . . ) are not an atomic property but arise fromthe current atomic surroundings 1

• OC “mechanisms” are highly simplified in many additional respects.

Animation: K + Br2

(courtesy of Prof. Dr. Arne Luchow, RWTH Aachen)

1 David B. Cook: “Quantum Chemistry – a unified approach”, Imperial College Press, 2008.

10


Molecules as viewed by molecular dynamics:

typical chemistry sketch:

What is “wrong” with that?

• molecules are not rigid but internally oscillating (except at T = 0 K)

• many molecular parts are less rigid than usually assumed

• there is no separation in stable molecules / unstable intermediates;this is only a question of time scale

• typical scale of molecular movements/vibrations:IR spectrum: 30–3500 cm−1 ⇒ 1 ps — 10 fs

• as we will see: on the fs time scale, most reactions are rare events !

Animation: brAB isomerization

(from Ole Carstensen’s Diploma thesis, Hartke group)

11


Modeling molecular movements simply:

non-mathematical model:

• molecule = collection of atoms

• atoms “sufficiently heavy” for classical mechanics

• between the atoms we have:

– attractive forces (bond formation, preferred conformations, etc.)

– repulsive forces (Pauli principle = closed electronic shells resistoverlapping, no “cold fusion”, etc.)

A little bit of maths (next page) provides us with classical molecular dynamics(MD):

• basic mechanical quantities: coordinates and momenta (velocities) of allatoms

• time proceeds in discrete steps (order of 1 fs)

• atom coordinates at next time step arise from

– atom coordinates of present (and earlier) time steps

– forces between atoms at present time step

MD provides us with a “trajectory”:

• coordinates and momenta of all atoms at later times t > 0,

• after having started from given initial conditions (coordinates and mo-menta of all atoms at time t = 0),

• for a molecular system of our choice (masses of all atoms).

From trajectories, we can calculate other measurable properties.

12


Mathematical derivation: From Newton to Verlet. . .

We want to obtain the trajectory x(t) of one particle with mass m in onespatial dimension x, from Newton’s second law F = ma. We have

v =dx

dt:= lim

∆t→0

∆x

∆t(6)

On a computer, this limit is problematic ⇒ finite-differences approximation:

v =dx

dt≈ ∆x

∆t=x1 − x0

t1 − t0(7)

for a discretized time coordinate, with abbreviation xn = x(tn) = x(t0+n×∆t).Similarly, we obtain

a =dv

dt≈ ∆v

∆t=v2 − v1

t2 − t1(8)

Using eq. 7, we can rewrite this to:

a =d2x

dt2≈

x2−x1

∆t −x1−x0

∆t

t2 − t1=x2 − 2x1 + x0

(∆t)2(9)

Inserting eq. 9 into F = ma yields after minor re-arrangement:

x2 = 2x1 − x0 +(∆t)2

mF1 (10)

or if iterated many times, for a general iteration step i→ i+ 1:

xi+1 = 2xi − xi−1 +(∆t)2

mFi (11)

This is the simple Verlet algorithm for calculating (time-discretized) classical-mechanical trajectories on computers. The generalization to 3D and to manyparticles simply amounts to writing down one copy of eq. 11 for each ofthe 3 coordinates of each particle.

The force Fi at coordinate value xi is obtained from the gradient of theinteratomic potential:

Fi = − dV

dx

∣∣∣∣x=xi

(12)

More on the “force field” F (~r) or V (~r) below.

13


How good is MD?

MD = numerical approximation tothe analytical solution of a differential equation

For a simpler, 1st-order differential equation

x′ =dx

dt= f(t, x) (13)

the finite-difference approximation of the derivative

dx

dt≈ ∆x

∆t(14)

results in∆x = xi+1 − xi = f(ti, xi) ∆t (15)

or slightly re-arranged:

xi+1 = xi + f(ti, xi) ∆t (16)

In graphical representation, we get this:

t0

x0

t1

t2

x

t

• all slopes f(ti, xi) (right-hand side of eq. 13) are exact ;

• but already the first time step (t1, x1) contains errors ,because finite-difference = no change in slope within the step;

• every following step accumulates errors from all previous steps.

14


How good is MD?

Consequences:

• accuracy depends on size of time step;

• time step can be

too large: finite-difference approximation (eq. 14) deteriorates;

too small: limited number n of digits⇒ total failure if change in position < 10−n × position

Useful accuracy in a window in-between these limits.

• force field dictates time step size (∆t ≈ 1fs).

• accuracy deteriorates with trajectory length:

– good accuracy for limited number of time steps (some ps total time)

– mathematically incorrect for many time steps (> some ns totaltime)

• most of the computational expense is in calculating forces;needed once per time step;⇒ influences balance accuracy–expense.

15


MD in practice

16


Force fields: functional form

Simplest possible (”‘class-I-”’) force field (actually: potential):

V (~r) =∑bonds

ki2

(`i − `i,0)2 (17)

+∑

angles

ki2

(ϑi − ϑi,0)2 (18)

+∑

torsions

kn2

(1 + cos(nω − γ)) (19)

+N∑i=1

N∑j>i

qiqj4πε0 rij

(20)

+N∑i=1

N∑j>i

4εij

[(σijrij

)12

−(σijrij

)6]

(21)

17


Force fields: functional form

”‘class II”’: couplings (”‘cross terms”’), z.B. stretch–torsion.

Further improvements:

• anharmonicity

• polarizability

• point-like charges → smeared charges

• reactive force fields

18


Force fields: parameters

Parameter values ← fitting MD results to reference data(experiments and/or ab-initio calculations)

Design decision:

universal system-specific

any atom in any molecule atom X in molecule Y in conformation Zmaximal generality maximal accuracy

compromise: atom types in many standard force fields;advantage: rather accurate, moderate expense;disadvantage: atom types fixed ↔ no chemical reactions!

Beispiel: OPLS-AA force field, 170 atom types for H,C,N,O;e.g. for O-atoms:

atom 16 7 OH OH Alcohol 8 15.999 2

atom 24 10 OH OH Trifluoroethanol 8 15.999 2

atom 29 10 OH OH Phenol 8 15.999 2

atom 31 10 OH OH Diols 8 15.999 2

atom 33 12 OS O Ether 8 15.999 2

atom 39 13 OS O Acetal 8 15.999 2

atom 40 10 OH OH Hemiacetal 8 15.999 2

atom 83 22 O NC=O Amide 8 15.999 1

atom 95 22 O O Urea 8 15.999 1

atom 99 22 O O=CNHC=O Imide 8 15.999 1

atom 108 26 OH OH Carboxylic Acid 8 15.999 2

atom 109 22 O C=O Carboxylic Acid 8 15.999 1

atom 112 28 O2 COO- Carboxylate 8 15.999 1

atom 118 22 O HC=O Aldehyde 8 15.999 1

atom 121 22 O C=O Ketone 8 15.999 1

atom 164 43 OW O Water (TIP3P) 8 15.999 2

atom 166 45 OW O Water (SPC) 8 15.999 2

atom 170 47 O- O- Hydroxide Ion 8 15.999 1

19


Force fields

typical “universal” force fields: MM2, MM3, MM4, OPLS, UFF, . . .for proteins: AMBER, CHARMMoligosaccharides: GLYCAMwater: SPC/E, TIP3P, TIP4P, . . .

attention: force fields cannot be mixed arbitrarily!

No MD simulation is better than the force field it uses.

20


Reactive force fields

Wrong prejudice 1:bond formation/dissociation is impossible with force fields.

Correct views:

• several force fields (previous page) have harmonic-oscillator stretch po-tential and immutable atom types ⇒ no reactions

• but there are other force fields: ReaxFF, COMB, EVB-QMDFF, Tersoff,Brenner, Finnis-Sinclair, Gupta,. . .anharmonic stretch potential, no atom types ⇒ reactions!

harmonicMorse

0

100

200

300

400

500

600

700

800

900

0 0.5 1 1.5 2 2.5 3 3.5 4

Epot

/ kJ/

mol

distance / Angstroem

21


Historically, reactive force fields were very normal:

• first MD calculation ever:B. J. Alder and T. E. Wainwright, J. Chem. Phys. 31 (1959) 459.

square well

dissociative

• early MD citation classic, on H+H2 → H2+H:M. Karplus, R. N. Porter and R. D. Sharma, J. Chem. Phys. 43 (1965)3259.

0 0.5

1 1.5

2 2.5

3

0

0.5

1

1.5

2

2.5

3

-600

-500

-400

-300

-200

-100

0

100

• introduction of the Verlet algorithm:L. Verlet, Phys. Rev. 159 (1967) 98.

Lennard−Jones

dissociative

22


Wrong prejudice 2: (can be found in many scientific papers)

1. bond formation and dissociation is determined by the electrons

2. electrons have to be treated by quantum mechanics (with orbitals)

3. and this cannot be done by a classical-mechanical force field.

Correct view:Items 1 and 2 are right, item 3 is nonsense!

• no force field tries to describe electrons explicitly

• force fields interpolate/approximate solutionsof the electronic Schrodinger equation

• there is NO link between force fields and classical mechanics.

23


Some technical details

• periodicity: “infinitely large” systems with finite simulation box, no“surface effects”; simulation of bulk solids, liquids, surfaces

• long-range forces (mostly Coulomb): cutoffs neighborhood lists, Ewaldsummation

• parallelization of the computationally intensive force calculation

• quality checks: energy, linear momentum, angular momentum constant?

• ensembles:

– microcanonical: NVE constant

– canonical: NVT, NPT

• phases of a typical MD calculation:

– system setup: atom types, initial coordinates & velocities

– energy minimization → eliminates worst “packing errors”

– equilibration: more realistic energy distribution, adjust velocities todesired temperature

– production; save trajectory “snapshots” every N steps

– analysis: calculate desired properties from snapshots

program packages: amber, charmm, NAMD/VMD, gromacs, LAMMPS,...

24


MD in practice

One single trajectory makes no sense:

• chaotic dynamics, but deterministic

• initial conditions not exactly known:

– scale problems:⇒ limited accuracy in preparation and measurement

– quantum mechanics:⇒ limited accuracy in preparation and measurement

– scale problem particle number: one molecule not interesting,molar numbers cannot be controlled individually.

Initial conditions are distributions.

⇒ larger sets of trajectories with distributed initial conditions,average results (with Boltzmann weights)

Number of trajectories depends on desired results:

• quantitative details: thousands–millions

• qualitative mechanisms: dozends–hundreds

convergence tests w.r.t. number of trajectories necessary!

25


MD in practice

Different view: ergodic theorem

ensemble average = time average

(r.h.s. also implies independence from initial conditions!)

Necessary condition: All points in phase space must be reached, in a finitenumber of steps.Impossible in practice: curse of dimensionality (exponential increase of phasespace with system size)

Ergodicity can be reached approximatively; then we have:

• one long trajectory replaces many short ones;

• erroneous drift of long trajectory away from true solution is irrelevant(it drifts from one initial condition to others)

• convergence checks w.r.t. trajectory length(typically needed: ps–ns = thousands–millions of time steps)

26


From trajectories to molecular physics/chemistry

typical calculated properties:

• (pair) distribution functions

• temperature, pressure, heat capacity

• spectra (e.g. IR from total dipole autocorrelation function)

• transport properties (e.g. diffusion coefficients, from the square of theaveraged path lengths or from velocity autocorrelation)

• entropy, free energy, k(T )

there are:

static equilibrium properties:completely time-independent average values

dynamical equilibrium properties: time-dependent properties correspond-ing to small perturbations; can be calculated from equilibrium properties(e.g. transport)

27


From trajectories to molecular physics/chemistry

two classes of properties

mechanical (or structural): depend on derivatives of the partition function,(Boltzmann-weighted) averages of simple functions of positions and mo-menta of all particles;e.g.: pair distribution functions, pressure, heat capacity

thermal: depend on the partition function itself (or on the volume ofaccessible phase space);e.g.: entropy, free energy, k(T )

⇔ massive difference in computational accessibility:

• mechanical properties: Boltzmann-weighted averages arise quasi auto-matically during MD; not problematic.

• thermal properties: all accessible areas of phase space must have beenvisited at least once (ergodicity); this is highly problematic for high-energy regions: combines curse of dimensionality with exp(E)-weights

28


Why force fields?

Alternative: ab-initio MD

• forces on atoms from gradient ∂E∂R , with electronic energy E from

quantum-chemical calculation of the electronic wavefunction

• classical-mechanical dynamics of the nuclei

With this, accessible today (J. Hutter, WIREs Comput. Mol. Sci. 2 (2012)604) on massively parallel high-performance hardware:

• 100–1000 atoms

• 2–200 ps simulated time in 1 week wallclock time

• time step ≈ 0.2 fs; ⇒ 104–106 time steps in 1 week wallclock time

• 1012 time steps would require 19230 years!

29


Why force fields?

Force-field MD:

• forces on atoms from gradient ∂E∂R , with approximation of the electronic

energy E by a “force field” (potential function)

• classical-mechanical dynamics of the nuclei

With this, accessible today (D. E. Shaw et al., Proc. Natl. Acad. Sci.USA 110 (2013) 5915; A. Nakano et al., Int. J. High Perf. Comput. Appl.22 (2008) 113.) on massively parallel high-performance hardware:

• extremely long simulated times, e.g. folding of ubiquitin: 8 ms(time step 5 fs ⇒ 1.6× 1012 time steps!,but wallclock time presumably . 1 year)

• huge systems: 1.34× 1011 atoms in simulation of materials deformationby hypervelocity impact

30


Reactions are rare events

With a barrier of 0.5 eV ≈ 50 kJ/mol and a typical value of the Arrheniuspre-factor, crossing the TS happens 1000 times per second. A direct MDsimulation of that would need 1012 steps (with one force calculation per step)and thus several 1000 years of CPU-time, just to cover the time betweentwo TS crossings.

But for reasonable “statistics” (reasonably small error when averaging overdifferent initial conditions) one needs several 100–1000 TS crossings.

abstract “movies” of rare events

Main conclusions:

• typical chemical energies only are sufficient to cross one (or very few)TS barriers, in one (or very few) “directions” ( = degrees of freedom,DOF)

• but almost always this energy is distributed roughly evenly over allDOFs ⇒ too little energy in the reactive DOF(s)

• re-distribution of the energy such that almost all of it is in the reactiveDOF(s) takes a long time. . .

• Macroscopically, we fail to notice that (time-scale discrepancy!);

• but microscopically (also in simulations!) this is a big problem.

possible tricks:

• combine (a) RRKM/TST-style statistics for the “path” from reactantminimum to TS with (b) trajectories started directly at the TS tocalculate the so-called “transmission coefficient” (TST correction)

• ”‘steered MD”’, etc.: introduction of artificial forces that push the systemacross the TS, combined with free MD in the other degress of freedom;then corrections to eliminate these artificial forces, or integration ofthese forces to obtain free energies.

31


Beyond MD

Monte-Carlo (MC): different access to statistical thermodynamics, via random“jumps” with a Metropolis acceptance criterion (no true dynamics)

continuum solution models:explicit solvent molecules → solvent as dielectric continuum

“coarse grained” force fields: join several atoms into “superatoms” and treatonly those.

dissipative particle dynamics (DPD): replace the “eliminated” degrees offreedom in coarse-grained force fields by additional forces: friction andrandom kicks.

Advantages of coarse-graining and DPD:

• fast but uninteresting movements (stretch vibrations of CH and OH)are hidden ⇒ longer time steps, longer total simulated times

• fewer explicitly treated particles ⇒ larger systems, smaller configurationspaces

32


33


MD vs. Quantum, part 1: pro-MD

Prejudices against classical mechanics & force fields:

• Prejudice: ”‘Quantum-mechanical results are so weird that they cannotbe modeled by force fields.”’

Reality: The lowest-energy solution of the electronic Schrodinger equationis a smooth, simple function of nuclear positions; can be modeled by aforce field. (Similarly for excited states.)

• Prejudice: ”‘force-field-MD cannot treat chemical reactions, because forcefields can neither treat (a) bond breaking/formation nor (b) changes inoxidation/valence states (sp2 → sp3 o.a.)”’

Reality: see above: force fields can be made reactive, by using Morse-likestretch potentials (instead of harmonic oscillators) and by not assigningfixed atom-types.

• Prejudice: ”‘Despite all that, quantum mechanics somehow is totallydifferent.”’

Reality: in chemically relevant situations, quantum-mechanical dynamicsof nuclei occasionally produces small quantitative corrections of classical-mechanical results, but rarely qualitative differences.

reactive scattering F+H2 classical vs. QM

True MD problems arise from scale discrepancies:

• huge particle numbers, long time intervals → coarse-graining

• entropy/free energy: advanced sampling techniques

• reactions = rare events → special tricks

34


QCT: Aoiz et al., Chem. Phys. Lett. 223 (1994) 215.

QM: Castillo et al., J. Chem. Phys. 104 (1996) 6531.

exp: Neumark et al., J. Chem. Phys. 82 (1985) 3045.

35


MD vs. Quantum, part 2: pro-Quantum

fundamental deficiencies of classical mechanics:

• no tunneling:

– potential well slightly too narrow

– potential barriers slightly too big

• particles point-like instead of ”‘smeared-out”’:

– barrier-crossing is “all or nothing”(quantum-mechanical: tunneling & above-barrier reflection)

• no zero-point energy: wrong energy distribution

– both statically

– and dynamically(vibrational energy redistribution, spectator modes);

– and again errors in barrier heights

• no interferences:

– between different trajectories on the same reaction path,

– or between different reaction paths.

⇒ quantum-mechanical treatment of nuclear dynamics: last chapter.

In chemistry, electrons are purely quantum-mechanical objects!⇒ need to be treated quantum-mechanically → ”‘Quantum Chemistry”’= next chapter. . .

36


Born-Oppenheimer separation/approximation:

Fact: Electrons are 2000–20000 times lighter than nuclei and hence movecorrespondingly faster = electrons and nuclei move on different time scales.

Consequence: It is physically reasonable and makes calculations easier touse a two-step picture:

1. separate treatment of the electrons, with nuclei fixed in space → electronicstates; i.e., the electronic problem is separated off and solved first.Solutions = electronic energies as functions of all nuclear coordinates= potential energy surfaces

2. solution of the whole problem reduces to calculating the nuclear dynamicson these (coupled) potential energy surfaces.

This Born-Oppenheimer separation still is exact !

-103.8

-103.75

-103.7

-103.65

-103.6

-103.55

-103.5

-103.45

-103.4

-103.35

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rtre

e

R / bohr

I(2P3/2)+CN(X

2Σ

+I(

2P1/2)+CN(X

2Σ

+I(

2P3/2)+CN(A

2Π

I(2P1/2)+CN(A

2Π

conical intersection

Born-Oppenheimer approximation:neglect the couplings between the potential energy surfaces⇒ nuclear dynamics on one single potential energy surface.

37


Basic quantum mechanics: Schrodinger equation

time-dependent, one spatial dimension x:

i~∂

∂tΨ(x, t) = HΨ(x, t) (22)

with Hamilton operator (Hamiltonian):

H := T + V := − ~2

2m

∂2

∂x2+ V (x) (23)

• classical mechanics: x(t) is the fundamental quantity (x depends on t);quantum mechanics: Ψ(x, t) is the fundamental quantity(x and t are in(!)dependent variables)

• Ψ(x, t) has “no direct” interpretation, all measurable quantities arecalculated from Ψ, as expectation values of the corresponding operators:

a =〈Ψ|A|Ψ〉〈Ψ|Ψ〉

(24)

special case: stationary states

ad-hoc separation ansatz:

Ψ(x, t) = ψ(x)f(t) (25)

yields the time-independent Schrodinger equation:

Hψ(x) = Eψ(x) (26)

• this separation ansatz works, but is not general!

• stationary state = all measurable quantities are time-independent.

• treating electronic states as stationary is justified, e.g. the electronicground state close to the equilibrium geometry and in absence of externalfields (laser).

38


Basic quantum mechanics: Hamiltonian

For every (arbitrary!) molecular system with electrons i, j = 1, 2, 3, . . . , nand nuclei A,B = 1, 2, 3, . . . ,M , we have this Hamiltonian:

H = −M∑A=1

1

2MA∇2A kinetic energy of the nuclei = TN (27)

−n∑i=1

1

2∇2i kinetic energy of the electrons (28)

+n∑i=1

n∑j>i

1

rijCoulomb repulsion between electrons (29)

−n∑i=1

M∑A=1

ZAriA

Coulomb attraction electrons–nuclei (30)

+M∑A=1

M∑B>A

ZAZBRAB

Coulomb repulsion between nuclei (31)

= TN + Hel + VNN (32)

In step 1 of the B.-O. separation, this reduces to:

Hel =

(−

n∑i=1

1

2∇2i −

n∑i=1

M∑A=1

ZAriA

)+

n∑i=1

n∑j>i

1

rij(33)

=∑i

hi +∑ij

hij (34)

• ab-initio quantum-chemistry programs use this very general Hel;hence they can be used for arbitrary chemical systems;

• after B.-O. separation, VNN is a trivial constant, which is usually includedin ab-initio quantum-chemistry programs;

• important:there are 1-electron operators hi and and 2-electron operators hij

39


Side note: other forces in chemistry

“Chemistry = quantum-mechanical electrostatics”

Other fundamental physical forces are not important!

e.g. gravitation:

A cluster of 100 gold atoms on the surface of the earth experiences agravitational force of about 3.2 · 10−10 pN (pico-Newton).

Typical chemical forces:

• calculated forces for cis-trans-photoisomerization of azobenzene:300–600 pN

• experimentally (AFM) deduced forces for isomerizations and bond dis-sociations: pN to nN

• maximal slope ( = force) in a standard Lennard-Jones potential for theargon dimer: about 5 pN

⇒ (on earth,) gravitational forces are 10 orders of magnitude(!) smallerthan the smallest chemical forces.

40


Quantum chemistry: overview

Solution of the time-independent Schrodinger equation for n electrons, forspace-fixed nuclei (B.O. separation):

HelΨ(~r1, ~r2, . . . , ~rn) = EelΨ(~r1, ~r2, . . . , ~rn) (35)

Antisymmetrized product ansatz (Slater determinant) for the wavefunction,with molecular orbitals ψi:

Ψ(~r1, ~r2) =1√2

∣∣∣∣ψ1(~r1) ψ2(~r1)ψ1(~r2) ψ2(~r2)

∣∣∣∣ (36)

⇒ effective 1-electron equations (Fock equations):

fψi =

(h1 +

n∑j=1

heff,112,j

)ψi = εiψi (37)

with averaged Coulomb repulsion between electrons i and j in h12,j. Calcula-tion of the yet unknown molecular orbitals ψi as Fock-operator eigenfunctionscorresponds to minimization of the total electronic energy Eel.

Numerical solution: Expansion of the orbitals ψi in basis functions χν:

|ψi〉 =K∑ν=1

Cνi|χν〉 (38)

The elements of the resulting Fock matrix contain many 1e and 2e integralsover basis functions and operators:

Fµν = hµν +Gµν = hµν +∑k

∑ρσ

CρkCσk {2(µν|ρσ)− (µσ|ρν)} (39)

with hµν =

∞∫−∞

χµ(r1)

(− 1

2∇2

1 −∑A

ZAr1A

)χν(r1) dr1 (40)

and (µν|ρσ) =

∞∫−∞

∞∫−∞

χ∗µ(r1)χ∗ρ(r2)

1

r12χν(r1)χσ(r2)dr1dr2 (41)

which can be calculated in advance, after selection of the molecular systemand of the basis functions χν. Determination of the molecular orbitals ψithen corresponds to determination of the MO expansion coefficients Cνi,by “diagonalization” of the Fock matrix. Since h12,j or Gµν depend on allother molecular orbitals ψj, this calculation has to be iteratively repeated(self-consistent field (SCF) cycles).

41


Basis function expansion: basics

Expand ψ(x) into a set of known basis functions {φi(x)} (basis set):

ψ(x) =∑i

ci φi(x) (42)

reminder: Taylor series

1

1 + x= 1− x+ x2 − x3 + x4 − x5 ± · · · ≈

n∑i=1

(−1)ixi (43)

0.5

1

1.5

2

2.5

3

3.5

4

-1.5 -1 -0.5 0 0.5 1 1.5

1/(x+1)n=1n=2

n=10

Demo up to n=20

42


Basis function expansion: basics

reminder: Fourier series

| sin(x)| ≈ 2

π− 4

π

(1

22 − 1cos(2x) +

1

42 − 1cos(4x) +

1

62 − 1cos(6x)

)(44)

0

0.2

0.4

0.6

0.8

1

1.2

-3 -2 -1 0 1 2 3

Accuracy of a basis function expansion:

• for good accuracy, some/all basis functions should have all (necessary)properties of the target function;

• for a given number of basis functions:accuracy increases if basis functions are chosen that are more similarto the target function.

• with infinitely many basis functions, the expansion would be exact;this is practically impossible, but usually we have:

• “larger basis set = better approximation”(for linearly independent basis functions)

• basis-set error = remaining approximation error for finite basis;can be determined and made smaller by systematically increasing thebasis set size.

43


Solution of the Schrodinger equation

via expansion in basis functions:

In operator/function form, the Schrodinger equation is:

Hψ = Eψ (45)

By expanding ψ into a finite number of user-selected basis functions {φi}

ψ =n∑i=1

ciφi (46)

it can be transformed into a matrix-vector eigenvalue equation:

Insertion of expansion eq. 46 into eq. 45 yields:∑i

ciHφi = E∑i

ciφi (47)

After multiplication with 〈φj| from the left, we get:∑i

ci〈φj|H|φi〉 = E∑i

ci〈φj|φi〉 (48)

For an orthonormal basis {φi}, the overlap matrix is the unit matrix:

Sji = 〈φj|φi〉 = δij (49)

and the sum on the r.h.s. of eq. 48 collapses to a single term:∑i

ci〈φj|H|φi〉 =∑i

ciHji = Ecj (50)

There is one such equation for each j; we can collect all of them into onematrix-vector equation:

H~c = E~c (51)

with Hij being the elements of the matrix H and ci the elements of thevector ~c.

44


Example: 1D Morse oscillator in a sine basis

H = p/2µ+ V (x), with V (x) = De{1− exp[−β(x− xe)]}2;Parameters in atomic units: De = 5.0, β = 2.0, xe = 1.5, µ = 0.05 amu ≈ 91.144

n numerical analytical1 0.3257483257557883 0.32574833036450432 0.9443301620270594 0.94433017585320303 1.519025577977924 1.5190256010214884 2.049834573608373 2.0498346058693615 2.536757148918450 2.5367571903968206 2.979793303908249 2.9797933546038667 3.378943038577837 3.3789430984904988 3.734206352927147 3.7342064220567189 4.045583246955985 4.04558332530252410 4.313073720748337 4.31307380822791711 4.536677851722307 4.53667787083289712 4.716420262499438 4.71639551311746413 4.854119814635596 4.85222673508161714 4.970424726778137 4.94417153672535615 5.102696896182915 4.992229918048683

1

2

3

4

5

6

7

8

9

10

11

0 1 2 3 4 5

-2

-1

0

1

2

3

4

5

6

7

0 1 2 3 4 5

45


Example: 1D Morse oscillator in distributed Gaussians

H = p/2µ+ V (x), with V (x) = De{1− exp[−β(x− xe)]}2;Parameters in atomic units: De = 5.0, β = 2.0, xe = 1.5, µ = 0.05 amu ≈ 91.144

n numerical analytical1 0.3257483281296143 0.32574833036450432 0.9443301695380278 0.94433017585320303 1.519025592324165 1.5190256010214884 2.049834599285713 2.0498346058693615 2.536757195227862 2.5367571903968206 2.979793386442639 2.9797933546038667 3.378943178632529 3.3789430984904988 3.734206573395122 3.7342064220567189 4.045583564466534 4.04558332530252410 4.313074135778425 4.31307380822791711 4.536678273982599 4.53667787083289712 4.716401680901365 4.71639551311746413 4.852983810524478 4.85222673508161714 4.960405968478248 4.94417153672535615 5.076871557803291 4.992229918048683

-1

0

1

2

3

4

5

6

7

0 1 2 3 4 5

0

10

20

30

40

50

60

70

80

90

100

0 1 2 3 4 5

46


alternative view: linear variation problem

We want to minimize the energy

E = 〈ψ|H|ψ〉 (52)

by linear variation of an initially guessed wavefunction ψ, i.e., by varyingthe expansion coefficients ci in eq. 46. As constraint, ψ should remainnormalized:

〈ψ|ψ〉 =∑ij

cicj〈φi|φj〉 =∑i

c2i = 1 (53)

where we have assumed an orthonormal basis 〈φi|φj〉 = δij.

This constrained minimization can be performed with the method of Lagrangemultipliers. We then minimize the Lagrange function

L(ci, E) = 〈ψ|H|ψ〉 − E(〈ψ|ψ〉 − 1) (54)

=∑ij

cicjHij − E

(∑i

c2i − 1

)(55)

Setting the first derivative to zero yields:

∂L∂ck

= 0 =∑j

cjHkj +∑i

ciHik − 2Eck k = 1, 2, . . . , n (56)

Since H is hermitian, H is symmetric (Hki = Hik), and hence we have:

2∑j

cjHkj − 2Eck = 0 (57)

which simply is eq. 50 again. Hence, solving the matrix eigenvalue problemeq. 51 optimizes the basis-expanded wavefunction ψ such that the energy E

of ψ is minimized.

Additionally, one can show that such an approximated E is an upper boundon the true energy (cf. next page).

47


Footnote: variational principle

For the exact eigenfunctions ψi of H we have

Hψi = Eiψi (58)

and the functions ψi are a complete basis (since H is hermitian). Therefore,they can be used to expand an arbitrary function φ:

φ =∑i

ciψi (59)

The energy expectation value

E =〈φ|H|φ〉〈φ|φ〉

(60)

for φ then becomes:

E =

∑ij c∗i cj〈ψi|H|ψj〉∑

ij c∗i cj〈ψi|ψj〉

(61)

Using eq. 58 and the fact that the functions ψi are orthonormal or can bechosen to be orthornomal (again because H is hermitian), we get:

E =

∑ij c∗i cjEj〈ψi|ψj〉∑

ij c∗i cj〈ψi|ψj〉

(62)

=

∑ij c∗i cjEjδij∑

ij c∗i cjδij

(63)

=

∑i |ci|2Ei∑i |ci|2

(64)

≥ E0

∑i |ci|2∑i |ci|2

= E0 (65)

with the last line simply following from a straightforward ordering of theeigenvalues: E0 ≤ E1 ≤ E2. . .

Thus, for any arbitrary wavefunction φ the energy is always greater thanthe exact ground-state energy E0, and a smaller energy difference indicatesa better wavefunction φ. The equality is valid only if φ is the exact ground-state wavefunction; then all expansion coefficients ci in eq. 59 are zero withthe exception of c0 = 1.

48


important observations/consequences:

• The original operator/function problem (differential equation) has beentransformed into a simple, standard numerical problem.

• This eigenvalue problem is solved in two characteristic steps:

1. calculation of matrix elements Hij = 〈φi|H|φj〉(definite integrals → (real or complex) numbers);

2. solution of the matrix eigenvalue problem (“diagonalization”)

• both steps cost computer time, and scale differently:

1. numerical integration, scales here like n4

2. matrix diagonalization scales like n3.

• For a basis of n functions, we obtain a n×n matrix H and hence exactlyn eigenvalues (some of them can be degenerate). These eigenvalues areupper bounds to the true energies.

• The n corresponding (approximative) eigenfunctions are calculated viaeq. 46 from the n elements of the eigenvectors ~c.

• The basis function expansion of ψ is optimized to produce a minimalenergy. It does not follow that this ψ is also optimal to calculate other(molecular) properties.

49


Wavefunctions for electrons

A wavefunction for one single electron (w/o spin) is called orbital.

Such an orbital ψ can be calculated for any 1-electron Hamiltonian h1:

h1ψ = εψ (66)

The corresponding eigenvalue is the orbital energy.However, this would be stupid: valid only for 1-electron systems.But we want multi-electron molecules. . .

How could a many-electron wavefunction look like?

Simplest idea: Hartree product of 1-electron orbitals.

Assume that for electron #1 with coordinates ~r1 = (x1, y1, z1) and electron#2 with coordinates ~r2 we already have two orbitals ψ1 and ψ2. Then ourHartree product is:

ΨHPI (~r1, ~r2) = ψ1(~r1)ψ2(~r2) (67)

From where do we get the orbitals, w/o resorting to the unrealistic 1-electronsituation of eq. 66? See Hartree-Fock below. . .

50


Fundamental defects of the Hartree product

• in ΨHP (~r1, ~r2), the electrons are uncorrelated (independent from eachother), since the total probability density is a product of single-electronprobability densities:

|ΨHP (~r1, ~r2)|2 = |ψ1(~r1)|2|ψ1(~r2)|2 (68)

Highly simplified example:Probability P (rS) for a red square:

uncorrelated:P (rQ) = P (r)P (Q) = 1

212 = 1

4

correlated:P (rQ) = 1

2 6= P (r)P (Q) = 12

12

⇒ one single Hartree product is too simple. A sum of products wouldbe better (cf. CI below)

• surprisingly, we are not allowed to put numbers on electrons.

51


“Exotic” properties of electrons

• every electron has a “spin” of 1/2. This cannot be visualized easily.Corresponds to two possible values (“up”, “down”) or to two abstractspin functions α and β that are multiplied with a spatial orbital ψ togenerate “spin orbitals” ψα and ψβ. We pretend to ignore this here,but actually it has to be included.

• The electron is an elementary particle; we cannot differentiate betweentwo elementary particles of the same kind ⇒ numbering electrons iswrong. Or: If we do number them, we have to repair this mistake.

• Hel is invariant w.r.t. changes in electron numbering.

• From this, it follows directly (next page):

Ψ(~r2, ~r1) = ±Ψ(~r1, ~r2) (69)

This feature does not change over time but is a fundamental particleproperty: (+ : bosons, - : fermions). Electrons are fermions (spin 1/2),hence we have for electrons:

Ψ(~r2, ~r1) = −Ψ(~r1, ~r2) (70)

(or more generally: “-” for an odd number of numbering permutations,“+” for an even number).The many-electron wavefunction has to be antisymmetric w.r.t. changesin electron numbering.

52


Permutation and Symmetry/Antisymmetry

The operator P permutes the numbers of two particles.

The statement that the Hamiltonian H is invariant w.r.t. such a permutationis equivalent to the statement that H und P commute:

[H, P ] := HP − P H = 0 (71)

Commuting operators have a common set of eigenfunctions; in particular,each eigenfunction Ψ of H also has to be eigenfunction of P , but for anothereigenvalue c:

PΨ = cΨ (72)

Because of fundamental properties of P (see below) we have: c = ±1. Thuseq. 72 is identical to eq. 69.

Acting with P from the left on eq. 72 yields:

P PΨ = P cΨ = cPΨ (73)

(where we have used that the scalar constant c is not changed by theoperator P ). But obviously we have:

P P = 1 (74)

since applying the same permutation twice is identical to doing nothing.This we use on the LHS of eq. 73. Simultaneously, on the RHS of eq. 73we replace PΨ with cΨ (which is nothing but eq. 72 again). This produces:

Ψ = c2Ψ (75)

which can be correct only if:c2 = 1 (76)

We accept w/o proof that P is hermitian; then all of its eigenvalues arereal. Hence we have:

c = ±1 (77)

53


Slaterdeterminante

Improve the Hartree product:

• take indistinguishability into account: construct an “average” over allpossible permutations of particle numbering:

Ψ(~r1, ~r2) =1√2

(ψ1(~r1)ψ2(~r2) + ψ2(~r1)ψ1(~r2)) (78)

• take the fermionic character into account (antisymmetrization):

Ψ(~r1, ~r2) =1√2

(ψ1(~r1)ψ2(~r2)− ψ2(~r1)ψ1(~r2)

)(79)

Eq. 79 can be written more simply as a determinant:

Ψ(~r1, ~r2) =1√2

∣∣∣∣ψ1(~r1) ψ2(~r1)ψ1(~r2) ψ2(~r2)

∣∣∣∣ (80)

and is called Slater determinant .

Additionally, we have to include spin (replace spatial orbitals by spin-orbitals).

ATTENTION:Even in the Slater determinant the electrons are still uncorrelated !

54


Consequences:

• a Slater determinant for 3 electrons including spin is:

Ψ(~r1, ~r2, ~r3) =1√3!

∣∣∣∣∣∣ψ1(~r1)α ψ1(~r1)β ψ2(~r1)αψ1(~r2)α ψ1(~r2)β ψ2(~r2)αψ1(~r3)α ψ1(~r3)β ψ2(~r3)α

∣∣∣∣∣∣ (81)

It would not have been possible

– to use ψ1 also in the rightmost column

– or to use α instead of β in the 2nd column.

In both cases the determinant would be zero! (two equal columns)

• from this, a lot follows:

– Pauli principle: One spin-orbital can be occupied with only oneelectron (or one spatial orbital with at most two electrons, withpaired spins: up, down). 2

– Aufbau principle to put electrons in orbitals

– periodic table of the elements

– and hence the whole diversity of chemistry!

– Fermi hole: two electrons with same spin automatically avoid eachother (see below)

– this leads to negative (energy-lowering) additional terms for electronswith parallel spin in the total energy: “exchange terms”

important: Neither the Fermi hole nor the exchange terms are related tothe Coulomb repulsion between electrons. There is no exchange force, in thesense that there is no corresponding term in the Hamiltonian H. Instead,this arises from intrinsic electron properties (or is an artifact of our electronnumbering).

2 In contrast to many text books, the Pauli principle is not a “mysterious additional requirement”!

55


Probability density |Ψ±(r1, r1)|2 as function of electron pair distance r12, forthe wavefunction

Ψ±(1, 2) =1√2{1s(1)2s(2)± 2s(1)1s(2)} (82)

(corresponds to the spatial part of the singlet (Ψ+) or triplet function (Ψ−)of the excited He atom).

The Fermi hole leads to greater electron-electron distances in the tripletstate ⇒ lower energy.

The “Fermi heap” of the spin-paired electrons in the singlet state leads tohigher Coulomb repulsion ⇒ higher energy.

56


Fundamental ideas of Hartree-Fock

• Ansatz for the many-electron wavefunction: one Slater determinant ΨSD

• determine all the orbitals in the Slater determinant simultaneously, byminimizing the total energy Eel = 〈ΨSD|Hel|ΨSD〉

• this integral contains the following terms:1-electronen operators:

〈ψi(~r1)ψj(~r2)|h1|ψi(~r1)ψj(~r2)〉 = 〈ψj(~r2)|ψj(~r2)〉︸︷︷︸=1

〈ψi(~r1)|h1|ψi(~r1)〉 (83)

2-electron operators:

〈ψi(~r1)ψj(~r2)|h12|ψi(~r1)ψj(~r2)〉 =

+∞∫∫−∞

ψ∗i (~r1)ψ∗j (~r2)h12ψi(~r1)ψj(~r2) d~r1d~r2

=

∞∫−∞

ψ∗i

∞∫−∞

ψ∗j h12ψj d~r2

︸︷︷︸

heff,112,j

ψi d~r1 (84)

Here, heff,112 is an effective 1-electron operator, averaged over the proba-

bility density of electron #2 in orbital j: mean-field-approximation

• it can be shown that this minimization is equivalent to determining theorbitals as eigenfunctions of the 1-electron Fock operator f :(

h1 +n∑j=1

heff,112,j

)︸︷︷︸

f

ψi = εiψi (85)

The resulting orbital energies (and the total energy) are upper boundsto the corresponding true values.

“Small” problem:To calculate ψi from eq. 85, we already need to know all other orbitals ψj,which we can only obtain from eq. 85, etc.

57


Way out: SCF procedure

orbitals

guess initialassemble

new Fock−

operator

solve Fock

equations

eigenfunctions

= new orbitals

old orbitals =

new orbitals?self−

consistent

solution

yes

no

• additional convergence criterion: the total energy should decrease atfirst (minimization) and then become roughly constant (convergence)

• convergence to true solutions is not guaranteed;oscillations between meaningless values is also possible.

• We need an initial guess of orbitals; this influences

– the convergence behavior (number of iterations);

– which of several possible solutions is found.

• possibilities to obtain starting values for the orbitals:

– neglect heff,112,j in the first step;

– use orbitals from another method w/o SCF cycles (e.g. semiempirics);

– results of a previous HF calculation with a smaller basis.

next: solution of the Fock equations by basis function expansion. . .

58


Basis functions for quantum chemistry:

Fundamental considerations:

• molecular electron distribution ≈ superposition of atomic ones

• electron-nuclear attraction ⇒ electron density maximal at nuclei.Example: electron density in ethene:

⇒ use basis functions that

• do not cover all space

• but are spatially compact

• quickly decay outwards

• and are centered at the nuclei(remember B.-O.: we prescribe the fixed nuclear coordinates!)

In crude approximation, each electron interacts witha nucleus shielded by the charge of all other electrons.

Situation similar to H-atom⇒ use H-atom-like basis functions.

59



Slater functions (Slater-type orbitals, STOs) are similar to simplified H-atomeigenfunctions:

χ(r, θ, φ) = rn−1e−ζr Ylm(θ, φ) (86)

Radial part, for n = 1, for two nuclei at long distance from each other:

Kern1

Kern2

Two essential changes upon bond formation (small internuclear distance)(Attention: both effects are graphically exaggerated!):

1. change in widths of the STOs:

this would require optimization of ζ, difficult since non-linear; replacedby linear(!) optimization of an STO-combination with fixed ζ values(“double zeta”, “triple zeta”, etc.):

χ(ζopt) = c1χ(ζ1) + c2χ(ζ2) + c3χ(ζ3) + · · · (87)

60


2. STOs become asymmetric (“polarization”):

represented as linear combination of different angular components (“po-larization functions” = χ-functions with greater quantum number l):

χ1s,pol = c1χ1s + c2χ2pσ (88)

61



Computational trick: use Gaussian functions (Gauss-type orbitals, GTOs)instead of STOs:

χGTO(x, y, z) = xiyjzke−αr2

(89)

(x-prefactors with exponents i, j, k “simulate” angular part Ylm).

GTOs are physically less accurate:

Slater Gauss

r = 0 cusp zero slope

r →∞ theoret. correct decay too fast

but:

• big error in absolute energies, smaller in relative energies

• “contraction” of GTOs to “simulate” STOs:

• calculation of integrals much easier.

⇒ almost all quantum-chemistry programs use GTOs.

62


Basis functions in real life

Automated distribution of basis functions to atoms, by selecting appropriatesets of functions from basis-function libraries. Examples:

Pople basis sets: e.g. 6-31G for a C-atom; corrsponds to a valence-double-zeta basis (VDZ); for today’s state of the art always too small (seebelow: polarization)

correlation-consistent basis sets: cc-pVnZ, with n=D,T,Q,5,6,. . .systematically adding groups of basis functions with equal importance⇒ systematic improvement of the results.

63


Basis functions in real life

Basis-set rules:

• more ζ values (VDZ, VTZ, VQZ, . . . ) increase radial flexibility, butthis is not enough, e.g.: in the limit of many s- and p-functions, NH3

is planar!⇒ higher angular functions (d,f,g,. . . ; polarization functions) increaseangular flexibility:e.g. DZP, pVDZ, 6-31G(d) = 6-31G∗, 6-31G(d,p) = 6-31G∗∗, . . .

• Basis set balance: artifacts can arise both from too few and fromtoo many polarization functions;rule of thumb: n . m−1, if m functions of one type and n next-higherangular functions are present. (e.g.: 4s 3p 2d 1f)

• anions and excited states have more diffuse electron densities ⇒ needto add spatially more extended basis functions:augmented, aug, 6-31+G, 6-31++G, . . .

• Attention: all basis sets mentioned so far actually are very small !e.g.: numbers of basis functions for H2O:

basis name number of basis functions

3-21G 136-31G∗ 19

cc-pVDZ 24cc-pVTZ 58cc-pVQZ 115cc-pV5Z 201cc-pV6Z 322

......

Typical density-functional calculations with plane-waves eikx as basisneed about 25000–100000 basis functions⇒ For STO or GTO basis sets, it is always necessary to test forconvergence w.r.t. basis set size!

64


Incoherent name warning:

LCAO = linear combination of atomic orbitals

is misleading:

• STO basis function correspond only approximately to atomic eigenfunc-tions;

• GTOs are even more different;

• it is a historically established but confusing custom to use for Ylm thesame “quantum numbers” l,m as for true atomic orbitals: 1s, 2s, 2p,3s, 3p, 3d, . . .However, there is no connection to the valence or hybridization stateof this atom in the molecule! E.g., a C-atom always needs “higherangular functions” (d,f,g) in the basis than what would correspond toits hybridization state (only s and p).

65


Basis functions: outlook

CBS: complete basis set extrapolation;extrapolation to the practically inaccessible limit of a complete basis(with infinitely many basis functions); possible with useful accuracy forthe rather systematic cc-pVnZ bases.

BSSE: basis set superposition error ;basis functions located at molecular part A influence the representationof the electronic wavefunction at molecular part B ⇒ systematic errorsin, e.g., dissociation energies; for small bases this error can be as largeas weak, non-bonding interactions (van-der-Waals).Error disappears in the complete basis limit; for smaller, incompletebases it can be approximately corrected with additional “ghost” basisfunctions (counterpoise correction).

ECP: effective core potential ;Representation of chemically inactive core electrons with a potential-likeadditional term in the Hamiltonian; only outer or valence electrons arerepresented explicitly (with then fewer basis functions); important forheavy transition-metal elements and as a simple way to partly includerelativistic effects.

66


The LCAO Fock matrix

The initially unknown molecular orbitals ψi are expanded in known basisfunctions χν:

|ψi〉 =K∑ν=1

Cνi|χν〉 (90)

The basis functions typically are non-orthogonal:

Sµν = 〈χµ|χν〉 =

∞∫−∞

χµ(r1)χν(r1) dr1 6= δµν (91)

Inserting the basis expansion eq. 90 into the HF equation eq. 85 yields:

f∑ν

Cνi|χν〉 = εi∑ν

Cνi|χν〉 (92)

After multiplication with 〈χµ| from the left, we have:∑ν

Cνi〈χµ|f |χν〉 = εi∑ν

Cνi〈χµ|χν〉 (93)

which is equivalent to the following matrix equation:

FC = SCε (94)

Here, matrix ε is a diagonal matrix of the orbital energies.

According to the definition of the Fock operator, the Fock matrix F is:

Fµν = hµν +∑k

{2(µν|kk)− (µk|kν)} (95)

mit hµν =

∞∫−∞

χµ(r1)

(− 1

2∇2

1 −∑A

ZAr1A

)χν(r1) dr1 (96)

Again inserting the basis expansion for ψk produces this expression for theFock matrix in AO basis:

Fµν = hµν +Gµν = hµν +∑k

∑ρσ

CρkCσk {2(µν|ρσ)− (µσ|ρν)} (97)

mit (µν|ρσ) =

∞∫−∞

∞∫−∞

χ∗µ(r1)χ∗ρ(r2)

1

r12χν(r1)χσ(r2)dr1dr2 (98)

67


LCAO-HF-SCF calculation flow-chart:

rate Start−

koeffi−zienten

neue

Fock−Matrix

aufstellen

Eigenvektoren

= neue Koeff.

Eingabe:

Molekülstruktur,

Basissatz

berechne

Integrale

speichere

Integrale

ja

nein

Fockmatrix

diagonalisieren

selbst−

konsistente

Lösung

alte Koeff. =

neue Koeff.?

E = E ?neualt

Resulting hardware considerations:

• in every SCF cycle, Fock matrix re-built from coefficients and integrals:

– huge integral file has to be written once and read n times;

– can be discarded after the calculation.

⇒ fast, local, huge but temporary disk space needed.

• permanent disk space almost zero.

• matrix diagonalization cannot be parallelized efficiently⇒ few CPU(-cores).

• many integrals and matrix elements to be stored in memory⇒ big RAM needed.

68


Results of HF:

• total electronic energy

• molecular orbitals (MOs); example: canonical HF orbitals for H2O:

2

5

24

69


Attention:

• Single MOs almost never have any concrete physical meaning!Physically relevant is the full many-electron wavefunction (built fromthe MOs via the Slater determinant) and observables calculated fromit (including the total electron density)

• HF MOs can be linearly combined arbitrarily, to new MOs; this doesnot change the HF energy or any other properties; e.g.: localization:

G. Knizia, Angew. Chem. 127 (2015) 5609 (DOI: 10.1002/ange.201410637)and J. Chem. Theory Comput. 9 (2013) 4834:

70


Results/reliability of HF:

Errors in Theoretical Chemistry:

For a given level of treating electronic correlation,differentiate basis set error ↔ method error (intrinsic error)

71


intrinsic geometry error of HF:

for bond lengths: approx. 3 pm (0.03 A) (≤ 3% error!)for bond angles: approx. 1.6 degrees

For comparison, experimental errors:for bond lengths: approx. 0.3 pm (0.003 A)for bond angles: approx. 0.2–0.5 degrees

(note: better methods like CCSD(T) (see below) have intrinsic errors of thesame size as experimental errors)

typical HF energy results for H2O:

Energy as function of bond angle (w/o dissociation) rather good in HF:

72


typical HF energy results for H2O:

Energy as function of bond length is o.k. near equilibrium distance, but forclosed-shell-HF is necessarily wrong in the dissociation limit (reason: seebelow!). Experimental value: 125.9 kcal/mol

73


Error compensation

Look out for:Method and basis set error can (partly) compensate each other!

typical example: geometry of H2O.Experimental values: ROH = 0.9578 A, αHOH = 104.48◦

Diagnosis:

• more/better basis functions ⇒ allow electrons to better avoid each other⇒ shorter bond;

• better electron-correlation treatment ⇒ electrons experience more e-e-repulsion ⇒ longer bond.

⇒ Errors from deficient electron correlation treatment (in HF) and fromtoo few basis functions can compensate each other. . .

Dangerous situation:

• improvements (larger basis, better correlation treatment) surprisinglygive worse results;

• error compensation is unsystematic = not reliable, not portable.

⇒ better: right results for the right reasons!

74


Is HF good or bad?

pro HF:

• gives up to 99% of absolute total molecular energies;

• finds the best possible 1-Slater-determinant wavefunction(within the provided basis set)

• in many situations, this “HF Slater determinant” has a weight of >90%in a better wavefunction (obtained by better methods)

• including the Fermionic character (indistinguishability, antisymmetry,spin 1/2) apparently is more important than correlation (for manysituation of chemical interest)

contra HF:

• 1% in absolute energies is not small compared to chemical effects;

• up to 25% of a covalent bond energy (e.g. in the H2 molecule) canresult from electron correlation;

• HF is systematically/qualitatively wrong in some situations, e.g. bonddissociation, bond lengths, frequencies, etc. (see below)

75


The potential-energy surface (PES)

Reaktions−koordinate

"wahre PES"

ProduktEdukt

Inter−mediat

PES gute Methode

PES schlechte Methode

En

erg

ie

• PES minimum = stable molecule

• PES saddle point (1st order) = reaction barrier

• on a good theory level, PES has qualitatively correct form

• on a better level, coordinates of stationary points almost identical, butimproved relative energies

• on a bad level, there are errors in:

– relative energies of stationary points,

– coordinates (location) of stationary points,

– number/type of stationary points.

On every level not advisable:using experimental molecular structures for reactants/products.

Better: find the true stationary points on this level of theory.

76


The energy gradient

PES = E(~R), potential energy E as function of nuclear coordinates ~R.

1st derivative (gradient) ∂E

∂ ~Rimportant for finding stationary points, see below.

numerical gradient: replace ∂E∂R with ∆E

∆R

• needs only energies, hence

• always possible (for every method to obtain E(~R)),

• but expensive: at least 1 (better 2,3,. . . ) energy calculations perdegree of freedom (DOF): example:

Aniline; suppose that one HF-SCF energy calculation takes 5 min;with 14∗3=42 DOFs and central differences (2 energies per DOF),the complete gradient vector needs 42∗2∗5 min = 420 min = 7 h(⇒ a converged geometry optimization needs � 1 day!)

analytical gradient: ∂E∂R is partially done analytically (only a small part

numerically)

• the SCF cycle is completed only once, then the analytical gradientis calculated w/o SCF, hence:

• calculating the complete gradient vector needs only 1–3 times asmuch time as the energy itself;

• but the actual geometry optimization still is numerical (and itera-tive).

in real life: geometry optimization only with methods for which an analyticalgradient is implemented! (else change to a different program package, or toa different level of theory)

77


Methods for energy minimization

steepest descent: each iteration = 1D minimization (line search) along thenegative gradient direction (strongest E decrease)

advantage: ensures E decrease in each iteration.

disadvantage: line search expensive; step directions orthogonal⇒ manysteps needed.

conjugate gradient: better choice of succesive search directions

advantage: arrives at minimum of a n-dimensional quadratic form in atmost n steps, qudratic convergence, avoids 2nd derivative (Hessianmatrix)

disadvantage: does not use 2nd-derivative information.

quasi-Newton: as conjugate gradient, but with iterative build-up of an(approximate) Hessian.

Attention: all methods converge to any stationary point, including e.g. saddlepoints ⇒ additional check (frequencies) necessary to establish nature of finalstationary point.

78


Global geometry optimization

All above geometry optimization methods are local : convergence always tothe minimum “closest” to the starting geometry.

Not problematic for traditional (in)organic molecules:

• guess the probable minimum-gemetry (“chemical intuition”)

• perhaps pre-optimization with a force field or with semiempirics

• local optimization at the desired level of theory

• if resulting geometry wrong, re-do from start. . .

• for resulting geometry, perhaps re-calculate energy at higher level oftheory

Insufficient for unconventional systems, e.g., clusters, protein folding, molec-ular docking, surface reconstructions:

• chemical intuition frequently fails;

• number of local minima increases exponentially with system size(Ar100 has approx. 1040 local minima!)

Ar250 Na+(H2O)18

⇒ global optimization methods:

• simulated annealing (SA)

• evolutionary algorithms (EA)

• basin-hopping

• . . .

79


Finding transition states (TS)

Strategy:

1. crude approximation to the TS

2. exact localization of the TS

3. confirmation as 1st-order saddle point (frequency calculation, see below)

techniques for crude approximation:

LST: (linear synchronous transit)linear interpolation between reactant and product structures,find maximal energy on this line.

QST: (quadratic synchronous transit)relax LST structure orthogonal to the LST path → structure X,quadratic interpolation between reactants, structure X, and products.find maximal energy on this quadratic connection.

80


mode-following, dimer method, etc.: several algorithms to follow a “pathof weakest ascent”; but this need not always lead to the desired TS:

exact TS localization: with standard geometry-optimization methods, but:

• at first, calculate a complete Hessian matrix;

• and then use an algorithm that ensures conservation of the correctnumber (1) of negative Hessian eigenvalues.

81


reaction path:

method to determine reaction paths:

1. start at the TS;

2. displace the vibrational mode with imaginary frequencyby a small step in positive and negative direction;

3. follow both descent directions (similar to an energy minimzation),until a minimum is reached.

applications:

• A TS found from known reactant and product structures does not haveto connect these two minima! Calculating the reaction path is a goodtest for that.

• Several methods can deduce dynamic-like properties from a 1D reactionpath. However, this is not a replacement for a full-dimensional dynamicscalculation.

82


Frequencies

2nd derivative of the energy with respect to nuclear coordinates (Hessianmatrix)

numeric ↔ analytic 2nd derivatives:as for the gradient, but even more important, b/c gradient has n components,Hessian matrix has n× n.

Normal coordinate analysis: for a molecule with n atoms

• only at stationary points (minima, saddle points, maxima)(gradient has to be zero)

• quadratic approximation to the potential⇒ good approximation only in small neighborhood of stationary point⇒ only for small displacements (small vibrational energy)

• diagonalization of the (mass-weighted) Hessian corresponds to a modelof 3n uncoupled harmonic oscillators, yielding:

– 3n eigenvalues and corresponding eigenvectors

– eigenvector = normal coordinate = linear combination of the carte-sian coordinates of all atoms = approximation to real molecularvibrations

– square root of an eigenvalue = frequency of the normal vibration

• 6 (for linear molecules 5) eigenvalues/-vectors correspond to translationand rotation of the whole molecule;corresponding frequencies should be exactly zero, but can attain valuesup to 50 cm−1 due to numerical errors ⇒ can be confused with true,low-frequency molecular vibrations; quantum-chemistry programs try toidentify these special “frequencies” automatically.

Attention: true molecular vibrations are more or less anharmonic and coupled!

83


Applications of frequencies

• characterization of stationary points:

minimum: all eigenvalues positive (all frequencies real)

1st-order saddle point: exactly one negative eigenvalues(exactly one imaginary frequency)

• zero-point energy correction (rectants, TS, products)

• → vibrational partition function → thermodynamic properties(only within one potential minimum ↔ MD)

harmonic HF frequencies

• HF frequencies systematically to high

• better agreeement with scaling factors of approx. 0.9

anharmonic frequencies: several methods for calculation,but very expensive (and accuracy of < 1 cm−1 only for 4-6–atomic molecules)

84


simulation of spectra

• normal vibration frequencies = approx. line positions

• intensities from derivative of dipole moment w.r.t. nuclear coordinates

• line width from convolution with an instrument function.

IR spectrum of Benzofuroxan:

85


Koopmans’ theorem

According to a simple derivation, we seem to have:

• negative orbital energies of occupied orbitals = ionization potential

• negative orbital energies of virtual orbitals = electron affinity

For some IPs from some HF calculations this seems to be correct, but twomajor errors are present:

• the derivation uses identical orbitals for the neutral and the chargedsystem,

• electron correlation is absent from HF.

⇒ error compensation for IPs, error enhancement for EAs.

better: two separate HF calculations for the neutral and the charged sys-tem, then take the energy difference (∆SCF) ⇒ different orbitals, no errorcompensation.

even better: two separate electron correlation calculations. . .

86


further molecular properties

Many properties can be calculated in two different ways, e.g. the dipolemoment as expection value of the dipole operator or as 1st derivative of theenergy w.r.t. the electrical field:

µ = 〈Ψ|r|Ψ〉 = −∂E∂F

(99)

For the latter strategy, finite differences can be used if no analytical derivativeis available (“finite field”).

Further derivatives make more properties available(F: electric field, B: magnetic field, I: nuclear spin, R: nuclear coordinates):

87


Population analysis

Atomic partial charges and (approximative) bond orders are spatial propertiesof the (observable) total molecular electron distribution.

Fundamental problem: Partitioning space into atomic regions is arbitrary!

historical:

Mulliken: Partitions total electron density in contributions from single basisfunctions; distribution onto the atoms at which these basis functionsare centered.disadvantage: basis functions centered at atom A can overlap with basisfunctions at atom B (cf. BSSE!) ⇒ no convergence with larger basissets.

Lowdin: variant of Mulliken (for orthogonalized basis functions).Different results, same disadvantage.

88


modern:

ESP (electrostatic potential): point charges on the atoms are optimizedsuch that the electrostatic field (calculated from the total electronicwavefunction) around the molecule is reproduced.disadvantage: underdetermined (charges on inner atoms not unique).

AIM (atoms in molecules, Bader): determines atomic regions by “topologicalanalysis” of the total electron density.

NAO/NBO (natural atomic orbitals, natural bond orbitals): iterative blockdiagonalization of a density matrix ordered by atoms.

exemplary results:

In practice:

• use one of the modern methods;

• never mix/compare results from different methods!

89


Open-shell systems

RHF: restricted Hartree-Fock

ROHF: restricted open-shell Hartree-Fock

UHF: unrestricted Hartree-Fock

• states with unpaired electrons cannot be treated with RHF wavefunctions:

• somewhat better depiction of dissociation:

UHF corresponds to two (coupled) HF-SCF calculations, one for α electrons,the other for β electrons. ROHF is more complicated.

90


Density-functional theory (DFT)

Functional: not new! Already in HF the energy is a functional (of thewavefunction).

The new item is “density”: hope that there is an energy-functional not of thewavefunction but of the (electron) density; avoids “unnecessary” complicationsdue to calculating “too much” information:Wavefunction is high-dimensional (dimensionality depends on number N ofelectrons):

Ψ(x1x2 . . .xN) (100)

while the electron density is only 3-dimensional (independent of N):

ρ(r1) = N

∫· · ·∫|Ψ|2ds1dx2 · · · dxN (101)

Key insights:

• The total electronic energy including(!) all electron correlation can bewritten exactly(!) as a functional of the so-called 2nd-order densitymatrix (6-dimensional).

• 1st theorem of Hohenberg and Kohn: an energy functional of the 3-dimensional density is possible w/o contradictions — but nobody knowsits functional form.

⇒ hope: If we find a suitably correct density functional, quantum-chemicalcalculations (1) become much simpler and (2) contain full electron correlation.

Historical attempts (1927 and following years) to construct pure DFT w/o awavefunction, failed: Teller has shown in 1962 that the DFTs of that timedid not allow for bound molecules. Key problem: Guessed functionals forthe kinetic energy were bad.

Recent developments (since 2010) have made such an orbital-free DFT (OF-DFT) for some systems (solids of main group elements) possible (B. G. delRio, J. M. Dieterich and E. A. Carter, J. Chem. Theory Comput. 13 (2017)3684 and literature cited therein). It remains to be seen if an extension toother systems (e.g., transition metals) will succeed.

91


Kohn-Sham DFT

More successful: re-introduction of the 3N -dimensional(!) wavefunction, asSlater determinant of molecular orbitals.

advantage: exact kinetic-energy functional for this wavefunction is known,as long as independent electrons are assumed (as in HF);

disadvantage: computational expense at least as large as in HF.

Important: a KS-DFT calculation is almost identical to a HF calculation,except:

• HF “exchange” is omitted

• instead there is an “exchange-correlation functional” Exc,covering these issues:

1. exchange

2. correlation

3. correct error in the kinetic-energy functional(electrons are not independent but correlated)

92


Hierarchy of density functionals

1st rung: LDA = SVWN (local density approximation, Slater-Vosko/Wilk/Nusair): exchange and correlation density functionals for the homoge-neous electron gas in infinitesimal volume elements are weighted withthe actual (inhomogeneous) density and added up.

2nd rung: GGA (generalized gradient approximation): functional containsboth density itself and the density-gradient (start of a Taylor series);typical GGA functionals: BLYP (Becke’s exchange functional, and cor-relation functional by Lee, Yang and Parr), BP86 (Becke’s exchangefunctional, Perdew’s correlation functional)

3rd rung: meta-GGA higher density derivatives (Laplacian of the density),or orbital kinetic energy density τ);typical meta-GGA functionals: VSXC (VanVoorhis/Scuseria exchangeand correlation), TPSS (Tao/Perdew/Staroverov/Scuseria)

4th rung: hybrid (or hyper-GGA): “exact” HF exchange added in; typicalhybrid functional: B3LYP (weighted mix of LDA, BLYP and HF-exchange, with 3 empirical weight-parameters)

5th rung: double hybrid contains also virtual orbitals, or combinationwith MP2 calculation using DFT orbitals: B2PLYP, mPW2-PLYP, etc.

Very roughly, both quality of results and expense of calculations increases inthis order. But this is not generally applicable, depends strongly on whichmolecular system is calculated.

93


DFT results

• LDA:

– bond lengths off by approx. 2 pm (mostly too short, but X-H andC≡C too long)

– bond angles off by approx. 1.9◦

– bond energies systematically too large (overbinding)

• GGA:

– geometries marginally better than with LDA (frequently overcom-pensated)

– but better energies

• meta-GGA, hybrid functionals:

– geometries can be on CCSD(T) level

– energies with & 2 kcal/mol almost “chemically accurate”

94


DFT compared to other methods

95


96


DFT in real life

• choice of functional highly unclear

• difference HF↔DFT: numerical integration of Exc;standard accuracy sometimes not sufficient for geometry optimizationsand/or frequency calculations for “floppy” molecules

• basis set requirements similar to HF = much smaller than for ex-plicit electron correlation! (see below): (aug)-cc-pVTZ recommended forquantitative results, with larger bases only marginal improvements

• for transition-metal complexes DFT still is the only reasonable option(CC methods are too expensive, HF and MP2 frequently wrong)

• all traditional functionals have incorrect long-distance asymptote ⇒ van-der-Waals interactions systematically wrong in plain DFT.possible remedies: DFT-D (DFT plus empirical vdW term (force field)),or vdW functionals.

• electron self-interaction: is exactly zero in HF (canceled out by HFexchange); can lead to errors in relative energies of up to 17 kcal/molin DFT; can be corrected only approximatively

• DFT MOs differ from HF MOs, but are not better or worse

• with present-day functionals, dissociation in DFT is just as problematicas with HF:

• restricted closed-shell, restricted open-shell, unrestricted: as for HF

• Is DFT ab-initio or semiempirical?

97

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