Optimization

of

parameters

for

semiempirical

methods

V

Modification

of

NDDO

approximations

and

application

to

elements

Several

modifications

that

have

been

made

to

the

NDDO

core-core

interaction

term

and

to

the

method

of

parameter

optimization

are

described

These

changes

have

resulted

in

a

more

complete

parameter

optimization

called

which

has

in

turn

allowed

elements

to

be

parameterized

The

average

unsigned

error

-LRB-

AUE

-RRB-

between

calculated

and

reference

heats

of

formation

for

species

was

kcal

mol

For

the

subset

of

compounds

involving

only

the

elements

H

C

N

O

F

P

S

Cl

and

Br

the

AUE

was

kcal

mol

The

equivalent

AUE

for

other

methods

were

HF

and

kcal

mol

Several

long-standing

faults

in

and

have

been

corrected

and

significant

improvements

have

been

made

in

the

prediction

of

geometries

Introduction

Over

the

past

years

NDDO-type

-LSB-

-RSB-

semiempirical

methods

have

evolved

steadily

The

earliest

of

these

methods

was

MNDO

-LSB-

-RSB-

which

itself

was

a

major

advance

over

even

earlier

non-NDDO

methods

such

as

-LSB-

-RSB-

The

main

advantage

of

MNDO

over

earlier

methods

was

that

the

values

of

the

parameters

were

optimized

to

reproduce

molecular

rather

than

atomic

properties

When

it

first

appeared

MNDO

was

immediately

popular

because

of

its

increased

accuracy

but

with

the

passage

of

time

various

limitations

were

found

among

the

most

important

of

which

was

the

almost

total

absence

of

a

hydrogen

bond

As

hydrogen

bonding

is

essential

to

life

this

particular

fault

essentially

precluded

MNDO

being

used

in

modeling

biochemistry

In

an

attempt

-LSB-

-RSB-

was

made

to

improve

MNDO

by

adding

a

stabilizing

Gaussian

function

to

the

core-core

interaction

to

represent

the

hydrogen

bond

Despite

the

fact

that

this

was

an

over-simplification

of

a

very

complicated

phenomenon

the

overall

effect

was

similar

and

for

the

first

time

NDDO

methods

gave

a

good

albeit

limited

model

of

hydrogen

bonding

In

the

course

of

the

next

several

years

improvements

were

made

to

the

method

of

parameter

optimization

The

result

of

this

was

the

method

-LSB-

-RSB-

which

culminated

in

the

parameterization

of

all

the

elements

in

the

main

group

in

-LSB-

-RSB-

At

the

same

time

various

changes

to

the

original

set

of

approximations

used

in

MNDO

were

proposed

the

most

important

of

which

were

the

addition

of

d-orbitals

to

main-group

elements

-LSB-

-RSB-

and

the

introduction

of

diatomic

parameters

Work

started

on

the

transition

metals

and

parameters

for

some

of

these

have

been

reported

-LSB-

-RSB-

More

recently

parameter

sets

tailored

to

reproduce

specific

phenomena

such

as

the

binding

energy

of

nucleic

acid

base

pairs

-LSB-

-RSB-

iron

complex

catalyzed

hydrogen

abstraction

-LSB-

-RSB-

phosphatase-catalyzed

reaction

barriers

-LSB-

-RSB-

and

the

redox

properties

of

iron

containing

proteins

-LSB-

-RSB-

have

been

developed

Because

of

the

way

advances

in

NDDO

developments

occurred

in

terms

of

the

modifications

of

the

approximations

and

the

extensions

to

specific

elements

or

groups

of

elements

there

has

been

an

inevitable

lack

of

consistency

The

aim

of

the

current

work

was

three-fold

to

investigate

the

incorporation

of

some

of

the

reported

modifications

to

the

core-core

approximations

into

the

NDDO

methodology

to

carry

out

a

systematic

global

parameter

optimization

of

all

the

main

group

elements

with

emphasis

on

compounds

of

interest

in

biochemistry

and

to

extend

the

methodology

by

performing

a

restricted

optimization

of

parameters

for

the

transition

metals

This

resulted

in

the

development

of

a

new

method

consisting

of

the

final

set

of

approximations

used

and

the

optimized

parameters

This

method

will

be

referred

to

as

parametric

method

number

or

The

name

was

chosen

to

avoid

any

confusion

with

two

other

unpublished

methods

and

Theory

Despite

the

apparent

complexity

of

semiempirical

methods

there

are

only

three

possible

sources

of

error

reference

data

may

be

inaccurate

or

inadequate

the

set

of

approximations

may

include

unrealistic

assumptions

or

be

too

inflexible

and

the

parameter

optimization

process

may

be

incomplete

In

order

for

a

method

to

be

accurate

all

three

potential

sources

of

error

must

be

carefully

examined

and

where

faults

are

found

appropriate

corrective

action

taken

Reference

data

In

contrast

to

earlier

methods

in

which

reference

data

was

assembled

by

painstakingly

searching

the

original

literature

the

current

work

relies

heavily

on

the

large

compendia

of

data

that

have

been

developed

in

recent

years

The

most

important

of

these

are

the

WebBook

-LSB-

-RSB-

for

thermochemistry

and

the

Cambridge

Structural

Database

-LSB-

-RSB-

-LRB-

CSD

-RRB-

for

molecular

geometries

During

the

early

stages

of

the

current

work

consistency

checks

were

performed

to

ensure

that

erroneous

data

were

not

used

These

checks

revealed

many

cases

in

which

the

calculated

heats

of

formation

were

inconsistent

with

the

reference

heats

of

formation

reported

in

the

NIST

database

On

further

checking

many

of

these

reference

data

were

also

found

-LSB-

-RSB-

to

be

inconsistent

with

other

data

in

the

WebBook

In

those

cases

where

there

was

strong

evidence

of

error

in

the

reference

data

the

offending

data

were

deleted

and

the

webbook

updated

-LSB-

-RSB-

For

molecular

geometries

gas

phase

reference

data

are

preferred

but

in

many

instances

such

data

were

unavailable

and

recourse

was

made

to

condensed-phase

data

Provided

that

care

was

taken

to

exclude

those

species

whose

geometries

were

likely

to

be

significantly

distorted

by

crystal

forces

or

which

carried

a

large

formal

charge

condensed-phase

data

of

the

type

found

in

the

CSD

were

regarded

as

being

suitable

as

reference

data

Because

earlier

methods

used

only

a

limited

number

of

reference

data

most

of

the

cases

where

the

method

gave

bad

results

were

not

discovered

until

after

the

method

was

published

In

an

attempt

to

minimize

the

occurrence

of

such

unpleasant

surprises

the

set

of

reference

data

used

was

made

as

large

as

practical

To

this

end

where

there

was

a

dearth

or

even

a

complete

absence

of

experimental

reference

data

recourse

was

made

to

high

level

calculations

Thus

for

the

Group

VIII

elements

there

are

relatively

few

stable

compounds

and

the

main

phenomena

of

interest

involve

rare

gas

atoms

colliding

with

other

atoms

or

molecules

so

reference

data

representing

the

mechanics

of

rare

gas

atoms

colliding

with

other

atoms

was

generated

from

the

results

of

ab-initio

calculations

Additionally

there

is

an

almost

complete

lack

of

thermochemical

data

for

many

types

of

complexes

involving

transition

metals

so

augmenting

what

little

data

there

was

with

the

results

of

ab-initio

calculations

was

essential

Use

of

Ab-Initio

results

Ab-initio

calculations

provide

a

convenient

source

of

reference

data

for

this

work

extensive

use

has

been

made

of

results

of

Hartree

Fock

and

density

functional

-LSB-

-RSB-

methods

-LRB-

DFT

-RRB-

both

with

the

-LRB-

d

-RRB-

basis

set

for

elements

in

the

periodic

table

up

to

argon

For

systems

involving

heavier

elements

the

functional

-LSB-

-RSB-

was

used

with

the

DZVP

basis

set

Within

the

spectrum

of

ab-initio

methods

these

methods

are

not

particularly

accurate

many

methods

with

larger

basis

sets

and

with

post-Hartree-Fock

corrections

are

more

accurate

However

the

methods

used

in

this

work

were

chosen

because

they

were

regarded

as

robust

practical

methods

allowing

many

systems

to

be

modeled

in

a

reasonable

amount

of

time

a

condition

that

could

not

be

achieved

with

the

more

sophisticated

ab-initio

methods

Procedure

used

in

deriving

ΔHf

Reference

heats

of

formation

ΔHf

for

compounds

and

ions

of

elements

for

which

there

was

a

paucity

of

data

were

derived

from

DFT

total

energies

in

two

stages

In

the

first

stage

a

basic

set

of

well-behaved

compounds

for

which

reliable

reference

values

of

experimental

ΔHf

were

available

was

assembled

Only

compounds

containing

one

or

more

of

the

elements

H

C

N

O

F

P

S

Cl

Br

and

I

were

used

For

this

set

a

root-mean-square

fit

was

made

to

the

reference

ΔHf

using

the

calculated

total

energies

Etot

and

the

atom

counts

Thus

the

error

function

S

in

Eq

-LRB-

-RRB-

was

minimized

In

this

expression

the

Ci

are

constants

for

each

atom

of

type

i

and

the

ni

are

the

number

of

atoms

of

that

type

In

the

second

stage

the

contribution

to

the

total

energy

of

compounds

containing

element

X

arising

from

the

elements

in

the

first

stage

was

removed

using

the

coefficients

from

Equation

-LRB-

-RRB-

A

second

RMS

fit

was

then

performed

In

this

the

function

minimized

S

was

the

RMS

difference

between

the

reference

ΔHf

of

compound

X

and

the

values

predicted

from

the

DFT

energy

Eq

-LRB-

-RRB-

In

this

expression

the

only

unknown

is

the

multiplier

coefficient

Cx

After

solving

for

Cx

the

ΔHf

of

any

compound

of

X

could

then

be

predicted

as

soon

as

its

DFT

total

energy

was

evaluated

Training

set

reference

data

The

training

set

of

reference

data

used

was

considerably

larger

than

that

used

in

parameterizing

-LSB-

-RSB-

where

approximately

discrete

species

were

used

In

optimizing

the

parameters

for

somewhat

over

separate

species

were

used

of

which

about

were

well-behaved

stable

molecules

The

remainder

consisted

of

reference

data

that

were

tailored

to

help

define

the

values

of

individual

parameters

or

sets

of

parameters

Use

of

rules

in

parameter

optimization

Most

reference

data

can

be

expressed

as

simple

facts

Indeed

all

the

earlier

NDDO

methods

were

parameterized

using

precisely

four

types

of

reference

data

ΔHf

molecular

geometries

dipole

moments

and

ionization

potentials

During

the

development

of

however

the

use

of

other

types

of

reference

data

was

found

to

be

necessary

Because

of

their

behavior

these

new

data

are

best

described

as

rules

In

this

context

a

rule

can

therefore

be

regarded

as

a

reference

datum

that

is

a

function

of

one

or

more

other

data

To

illustrate

the

use

of

a

rule

consider

the

binding

energy

of

a

hydrogen

bond

in

the

water

dimer

By

default

the

weighting

factor

for

ΔHf

for

normal

compounds

is

kcal

mol

With

this

weighting

factor

average

unsigned

errors

in

the

predicted

ΔHf

of

the

order

of

kcal

mol

would

be

acceptable

particularly

as

the

spectrum

of

values

of

ΔHf

spans

several

hundreds

of

kilocalories

per

mole

However

the

binding

energy

of

a

hydrogen

bond

in

a

water

dimer

is

only

kcal

mol

To

have

an

average

unsigned

error

-LRB-

AUE

-RRB-

of

kcal

mol

in

the

prediction

of

hydrogen

bond

energies

would

render

such

a

method

almost

useless

for

modeling

such

phenomena

One

way

to

increase

the

importance

of

the

hydrogen

bond

in

water

would

be

to

increase

the

weight

for

the

ΔHf

of

the

water

molecule

kcal

mol

and

the

water

dimer

system

ca

kcal

mol

While

this

would

have

the

intended

effect

of

increasing

the

weight

of

the

hydrogen

bond

energy

it

would

also

have

the

undesired

effect

of

increasing

the

weight

of

the

ΔHf

of

water

An

alternative

would

be

to

express

the

ΔHf

of

the

water

dimer

in

terms

of

the

ΔHf

of

two

individual

water

molecules

The

difference

between

the

two

ΔHf

that

of

water

dimer

and

that

of

two

isolated

water

molecules

would

be

the

energy

of

the

hydrogen

bond

If

the

weight

assigned

to

this

quantity

were

then

increased

it

would

increase

the

weight

for

the

hydrogen

bond

energy

without

also

increasing

the

weight

for

the

ΔHf

of

water

Such

a

reference

datum

is

referred

to

here

as

a

rule

That

is

rules

relate

the

ΔHf

of

a

moiety

to

that

of

one

or

more

other

moieties

Thus

in

the

above

example

the

simple

reference

datum

H

representing

the

ΔHf

of

an

isolated

water

molecule

could

be

expressed

as

Using

a

rule-based

reference

datum

to

represent

the

strength

of

the

hydrogen

bond

and

giving

a

weight

of

to

the

hydrogen

bond

energy

the

ΔHf

of

the

water

dimer

would

then

be

defined

as

In

this

expression

was

the

calculated

ΔHf

in

kcal

mol

of

an

isolated

water

molecule

This

rule

could

be

interpreted

as

The

calculated

strength

of

the

hydrogen

bond

formed

when

two

water

molecules

form

the

dimer

should

be

kcal

mol

and

the

importance

should

be

times

that

of

ordinary

heats

of

formation

Rules

are

very

useful

in

defining

the

parameter

hypersurface

Examples

of

such

tailoring

are

as

follows

Correcting

qualitatively

incorrect

predictions

During

the

parameterization

of

transition

metals

some

systems

were

predicted

to

have

qualitatively

the

wrong

structure

For

example

-LSB-

-RSB-

was

initially

predicted

to

have

a

tetrahedral

structure

instead

of

the

geometry

observed

To

induce

the

parameters

to

change

so

as

to

make

the

geometry

more

stable

than

the

Td

geometry

a

rule

was

added

to

the

set

of

reference

data

for

copper

compounds

This

rule

was

constructed

using

the

results

of

calculations

on

-LSB-

-RSB-

First

the

total

energies

of

the

optimized

structure

and

that

of

the

structure

resulting

from

the

semiempirical

calculation

were

evaluated

The

difference

between

these

energies

was

then

used

in

constructing

the

rule

In

this

case

the

rule

was

that

The

ΔHf

of

the

geometry

predicted

by

the

faulty

semiempirical

method

should

be

n.n

kcal

mol

more

than

that

of

the

geometry

When

such

a

rule

was

included

in

the

parameter

optimization

with

an

appropriate

large

weight

any

tendency

of

the

parameters

to

predict

the

incorrect

geometry

resulted

in

a

large

contribution

to

the

error

function

That

is

with

the

new

rule

in

place

there

was

a

strong

disincentive

to

prediction

of

the

incorrect

structure

Usually

one

rule

was

sufficient

to

correct

most

qualitative

errors

but

for

a

few

complicated

structures

more

than

one

rule

was

needed

The

commonest

need

for

multiple

rules

occurred

when

initially

one

rule

was

used

to

correct

a

faulty

prediction

and

after

re-optimizing

the

parameters

the

geometry

optimized

to

a

new

structure

that

was

distinctly

different

from

either

the

correct

structure

or

the

incorrect

structure

covered

by

the

rule

When

that

happened

the

procedure

just

described

was

repeated

and

a

new

rule

added

to

the

set

of

reference

data

to

address

the

new

incorrect

structure

In

extreme

cases

several

such

rules

might

be

needed

each

one

defining

a

geometry

that

was

incorrect

and

should

therefore

be

avoided

Rare

gas

atoms

at

sub-equilibrium

distances

For

some

elements

specifically

those

of

Group

VIII

there

is

an

understandable

shortage

of

useful

experimental

reference

data

In

addition

most

simulations

involving

these

elements

are

likely

to

involve

a

rare-gas

atom

dynamically

interacting

with

another

atom

or

with

a

molecule

at

distances

significantly

less

than

the

equilibrium

distance

This

makes

determining

the

potential

energy

surface

at

sub-equilibrium

distances

important

As

with

hydrogen

bond

energies

the

energies

involved

in

this

domain

are

likely

to

be

in

the

order

of

a

few

kcal

mol

The

shape

of

the

potential

energy

surface

-LRB-

PES

-RRB-

can

readily

be

mapped

using

DFT

methods

By

selecting

two

or

three

representative

points

on

this

PES

reference

data

rules

can

be

constructed

that

describe

the

mechanical

properties

of

the

interactions

As

with

hydrogen

bonding

a

large

weight

can

be

assigned

to

these

rules

Use

of

rules

to

restrain

parameter

values

In

general

uncharged

atoms

that

are

separated

by

a

distance

sufficiently

large

so

that

all

overlaps

between

orbitals

on

the

two

atoms

are

vanishingly

small

will

not

interact

significantly

and

what

interaction

energy

exists

would

arise

from

VDW

terms

of

their

nature

these

are

mildly

stabilizing

Although

statements

of

this

type

are

obviously

true

when

they

are

expressed

as

rules

and

added

to

the

training

set

of

reference

data

they

can

help

define

the

parameter

values

For

a

pair

of

atoms

A

and

B

a

simple

diatomic

system

would

be

constructed

in

which

the

interatomic

separation

was

the

minimum

distance

at

which

any

overlaps

of

the

atomic

orbitals

would

still

be

insignificant

The

electronic

state

of

such

a

system

would

then

be

the

sum

of

the

states

of

the

two

isolated

atoms

Thus

if

both

A

and

B

were

silicon

then

since

the

ground

state

of

an

isolated

silicon

atom

is

a

triplet

the

combined

state

would

be

a

quintet

Because

the

two

atoms

do

not

interact

significantly

a

rule

could

then

be

constructed

that

said

The

energy

of

the

diatomic

system

is

equal

to

the

addition

of

energies

of

the

two

individual

systems

By

giving

this

rule

a

large

weight

any

tendency

of

the

method

to

generate

a

spurious

attraction

or

repulsion

between

the

atoms

would

be

prevented

Atomic

energy

levels

In

keeping

with

the

philosophy

that

a

large

amount

of

reference

data

should

be

used

in

the

parameter

optimization

spin-free

atomic

energy

levels

were

used

for

most

elements

The

exceptions

were

carbon

nitrogen

and

oxygen

where

there

were

enough

conventional

reference

data

that

the

addition

of

atomic

energy

levels

would

not

significantly

improve

the

definition

of

the

parameter

surface

NDDO

approximations

do

not

allow

for

spin-orbit

coupling

Therefore

spin-free

levels

were

needed

For

a

few

elements

there

were

insufficient

spin

states

to

allow

the

spin-free

energy

levels

to

be

calculated

For

all

the

remaining

elements

spin-free

energy

levels

were

calculated

In

Moore

's

compendia

-LSB-

-RSB-

of

atomic

energy

levels

observed

emission

spectra

were

used

in

determining

the

energy

levels

of

the

various

states

of

neutral

and

ionized

atoms

Most

of

these

energy

levels

were

characterized

by

three

quantum

numbers

the

spin

and

orbital

angular

momenta

and

the

J

or

spin-orbit

quantum

number

The

starting

point

for

determining

the

spin-free

atomic

energy

levels

for

a

given

element

consisted

of

identifying

each

complete

manifold

of

atomic

energy

levels

for

that

element

that

is

each

set

of

levels

split

by

spin-orbit

coupling

If

all

members

of

the

set

were

present

i.e.

all

energy

levels

from

L+S

to

L

S

then

the

weighted

barycenter

of

energy

could

be

calculated

The

spin-free

energy

level

E

was

derived

from

the

spin-split

levels

E

-LRB-

S

L

J

-RRB-

using

Eq

-LRB-

-RRB-

In

those

cases

where

the

ground

state

of

an

atom

was

itself

a

member

of

a

spin-split

manifold

the

barycenter

of

the

ground

state

manifold

was

calculated

and

used

in

re-defining

the

spin-free

ground

state

For

all

elements

except

tungsten

this

change

in

definition

was

benign

There

is

a

level

present

in

tungsten

that

is

located

only

kcal

mol

above

the

ground

state

This

puts

it

inside

the

manifold

which

has

a

barycenter

at

kcal

mol

The

effect

of

this

was

that

on

going

from

a

spin-split

to

a

spin-free

ground

state

the

ground

state

changed

from

or

to

or

and

the

state

now

became

an

excited

state

with

an

energy

of

kcal

mol

To

allow

for

this

a

corresponding

change

was

made

to

the

ground

state

configuration

in

the

definition

of

tungsten

Where

there

were

relatively

few

other

reference

data

the

singly-ionized

and

in

rare

cases

the

doubly-ionized

spin-free

states

were

also

evaluated

and

used

as

reference

data

Each

energy

level

contributed

one

reference

datum

to

the

training

set

Most

atoms

have

a

large

number

of

atomic

energy

levels

so

in

order

to

minimize

the

probability

that

a

level

might

be

incorrectly

assigned

each

level

was

labeled

with

three

quantum

numbers

the

total

spin

momentum

the

total

angular

momentum

and

the

principal

quantum

number

for

these

two

quantum

numbers

These

were

compared

with

the

corresponding

values

calculated

from

the

state

functions

Since

each

set

of

three

quantum

numbers

is

unique

the

potential

for

miss-assignment

was

minimized

In

rare

cases

particularly

during

the

early

stages

of

parameter

optimization

two

states

with

the

same

total

spin

and

angular

quantum

numbers

would

be

interchanged

with

the

result

that

the

calculated

principal

quantum

number

would

also

be

interchanged

All

such

cases

always

involved

the

ground

state

and

were

quickly

identified

and

corrected

Approximations

Most

of

the

approximations

used

in

are

identical

to

those

in

and

The

differences

are

Core-core

interactions

In

the

original

MNDO

set

of

approximations

two

changes

were

made

to

the

simple

point-charge

expression

for

the

core-core

repulsion

term

Beyond

about

five

Ångstroms

there

should

be

no

significant

interaction

of

two

neutral

atoms

However

in

MNDO

the

two-electron

two-center

integrals

and

the

electron-core

interactions

do

not

converge

to

the

exact

point

charge

expression

instead

they

are

always

slightly

smaller

To

prevent

there

being

a

small

net

repulsion

between

two

uncharged

atoms

the

core-core

expression

is

modified

by

the

exact

term

being

replaced

by

the

term

used

in

the

integrals

An

additional

term

is

needed

to

represent

the

increased

core-core

repulsion

at

small

distances

due

to

the

unpolarizable

core

These

two

changes

can

be

expressed

as

the

MNDO

core-core

repulsion

term

as

shown

in

Eq

-LRB-

-RRB-

This

approximation

works

well

for

most

main-group

elements

but

when

molybdenum

was

being

parameterized

Voityuk

-LSB-

-RSB-

found

that

the

errors

in

heats

of

formation

and

geometries

were

unacceptably

large

and

good

results

were

achieved

only

when

a

diatomic

term

was

added

to

the

core-core

approximation

as

shown

in

Eq

-LRB-

-RRB-

When

parameters

for

elements

of

Groups

IA

were

being

optimized

the

MNDO

approximation

to

the

core-core

expression

was

found

to

be

unsuitable

In

these

elements

there

is

only

one

valence

electron

so

the

core

charge

is

the

same

as

that

of

hydrogen

A

consequence

of

this

was

that

the

apparent

size

of

these

elements

was

also

approximately

that

of

a

hydrogen

atom

in

marked

contrast

with

observation

For

these

elements

diatomic

core-core

parameters

were

also

found

to

be

essential

Further

examination

showed

that

when

diatomic

parameters

were

used

there

was

always

an

increase

in

accuracy

therefore

in

the

current

work

Eq

-LRB-

-RRB-

was

replaced

systematically

by

Eq

-LRB-

-RRB-

As

the

interatomic

separation

increased

Voityuk

's

equation

converged

to

the

exact

point-charge

interaction

as

expected

However

for

rare

gas

interactions

an

increase

in

accuracy

was

found

when

the

rate

of

convergence

was

increased

by

the

addition

of

a

small

perturbation

Subsequently

the

perturbed

function

was

found

to

be

generally

beneficial

Because

of

this

the

general

form

of

the

core-core

interaction

used

in

is

that

given

in

Eq

-LRB-

-RRB-

At

normal

chemical

bonding

distances

Eqs

-LRB-

-RRB-

and

-LRB-

-RRB-

have

essentially

similar

behavior

but

at

distances

of

greater

than

about

Å

the

effect

of

the

perturbation

is

to

make

the

function

significantly

smaller

than

the

Voityuk

approximation

d-orbitals

on

main-group

elements

Thiel

and

Voityuk

have

shown

-LSB-

-RSB-

that

a

large

increase

in

accuracy

results

when

d-orbitals

are

added

to

main-group

elements

that

have

the

potential

to

be

hypervalent

During

preliminary

stages

of

this

work

d-orbitals

were

excluded

from

main-group

elements

and

the

parameters

were

optimized

This

work

was

then

repeated

but

with

d-orbitals

on

various

main-group

elements

The

results

were

in

accordance

with

Thiel

's

observation

the

accuracy

of

the

method

increased

significantly

Because

of

this

d-orbitals

were

added

to

several

main-group

elements

the

value

of

the

increased

accuracy

far

outweighs

the

extra

computational

cost

The

effect

of

the

addition

of

d-orbitals

was

fundamentally

different

between

main-group

elements

and

transition

metals

For

main-group

elements

the

effect

of

d-orbitals

is

merely

a

perturbation

to

a

large

degree

the

chemistry

of

these

elements

is

determined

by

the

s

and

p

atomic

orbitals

This

is

not

the

case

with

transition

metals

where

the

d-orbitals

are

of

paramount

importance

and

the

s

and

p

orbitals

are

of

only

very

minor

significance

In

recognition

of

the

importance

of

the

s

and

p

shells

in

main-group

chemistry

specific

parameters

are

used

for

the

five

one-center

two-electron

integrals

Conversely

for

the

transition

metals

the

values

of

these

integrals

are

derived

directly

from

the

internal

orbital

exponents

Unpolarizable

core

As

noted

earlier

the

NDDO

core-core

interaction

is

a

function

of

the

number

of

valence

electrons

For

elements

on

the

left

of

the

periodic

table

these

numbers

are

small

and

can

cause

the

elements

to

appear

to

be

too

small

This

was

part

of

the

rationale

behind

the

adoption

of

Voityuk

's

diatomic

core-core

parameters

However

even

the

Voityuk

approximation

failed

during

parameter

optimization

when

in

rare

cases

a

pair

of

atoms

would

approach

each

other

very

closely

Examination

of

these

catastrophes

indicated

that

the

cause

was

the

complete

neglect

of

the

unpolarizable

core

of

the

atoms

involved

To

allow

for

its

presence

the

core-core

interaction

for

all

element

pairs

was

modified

by

the

addition

of

a

simple

function

fAB

based

on

the

first

term

of

the

Lennard-Jones

potential

-LSB-

-RSB-

A

candidate

function

was

constructed

Eq

-LRB-

-RRB-

using

the

fact

that

to

a

first

approximation

the

size

of

an

atom

increases

as

the

third

power

of

its

atomic

number

The

value

of

c

was

set

to

this

being

the

best

compromise

between

the

requirements

that

the

function

should

have

a

vanishingly

small

value

at

normal

chemical

distances

That

is

under

normal

conditions

the

value

of

the

function

should

be

negligible

and

at

small

interatomic

separations

the

function

should

be

highly

repulsive

i.e.

that

it

should

represent

the

unpolarizable

core

Individual

core-core

corrections

For

a

small

number

of

diatomic

interactions

the

general

expression

for

the

core-core

interaction

was

modified

in

order

to

correct

a

specific

fault

Because

it

is

desirable

to

keep

the

methodology

as

simple

as

possible

modifications

of

the

approximations

were

made

only

after

determining

that

the

existing

approximations

were

inadequate

The

diatomic

specific

modifications

were

O

H

and

N

H

In

the

original

MNDO

formalism

the

general

core-core

interaction

Eq

-LRB-

-RRB-

was

replaced

in

the

cases

of

O

H

and

N

H

pairs

with

Eq

-LRB-

-RRB-

An

unintended

effect

of

this

change

was

that

at

distances

where

hydrogen-bonding

interactions

are

important

the

diatomic

contribution

to

the

ΔHf

is

greater

than

if

the

general

approximation

Eq

-LRB-

-RRB-

had

been

used

This

contributed

to

a

reduced

hydrogen-bonding

interaction

in

MNDO

and

was

a

contributor

to

the

need

for

modified

core-core

interactions

in

and

In

the

MNDO

core-core

approximation

is

replaced

by

Voityuk

's

diatomic

expression

but

even

with

that

modification

the

resulting

hydrogen

bond

interaction

energy

was

too

small

In

an

attempt

to

increase

it

the

Voityuk

approximation

was

replaced

by

Eq

-LRB-

-RRB-

At

normal

O

H

and

N

H

separations

approximately

Å

Eqs

-LRB-

-RRB-

and

-LRB-

-RRB-

have

similar

values

but

at

hydrogen

bonding

distances

Å

the

contribution

arising

from

the

exponential

term

is

significantly

reduced

resulting

in

a

corresponding

increased

hydrogen

bond

interaction

energy

C

C

After

optimizing

all

parameters

it

was

found

that

compounds

containing

yne

groups

C

C

were

predicted

to

be

too

stable

by

about

kcal

mol

per

yne

group

This

error

was

unique

to

compounds

with

extremely

short

C

C

distances

and

in

light

of

the

increased

emphasis

on

accurately

reproducing

the

properties

of

organic

compounds

the

C

C

core-core

term

was

perturbed

by

the

addition

of

a

repulsive

term

This

term

was

optimized

to

correct

the

error

in

the

yne

groups

and

to

have

a

negligible

effect

on

all

other

C

C

interactions

The

optimized

form

of

the

C

C

core-core

interaction

is

given

in

Eq

-LRB-

-RRB-

Si

O

During

testing

of

neutral

silicate

layers

of

the

type

found

in

talc

were

found

to

be

slightly

repulsive

instead

of

being

slightly

bound

An

attempt

was

made

to

correct

for

this

error

by

adding

a

weak

perturbation

to

the

Si

O

interaction

illustrated

by

Eq

-LRB-

-RRB-

Nitrogen

pyramidalization

Although

predicted

the

degree

of

pyramidalization

of

primary

amines

correctly

it

overestimated

the

pyramidalization

of

secondary

and

tertiary

amines

The

degree

of

pyramidalization

of

these

amines

was

decreased

by

adding

a

function

to

make

the

calculated

ΔHf

more

negative

as

the

nitrogen

became

more

planar

as

shown

in

Eq

-LRB-

-RRB-

In

this

equation

the

angle

ϕ

is

a

measure

of

the

non-planarity

of

the

nitrogen

environment

and

is

given

by

2π

minus

the

sum

of

the

three

contained

angles

about

the

nitrogen

atom

For

planar

secondary

and

tertiary

amines

this

correction

amounted

to

kcal

mol

per

nitrogen

atom

More

elements

The

NDDO

basis

sets

of

many

of

the

elements

parameterized

in

have

not

previously

been

described

For

all

elements

except

hydrogen

which

has

only

an

s

orbital

the

basis

set

consists

of

an

s

orbital

three

p

orbitals

and

for

most

elements

a

set

of

five

d

orbitals

Slater

atomic

orbitals

are

used

exclusively

these

are

of

form

Where

ξ

is

the

orbital

exponent

n

is

the

principal

quantum

number

-LRB-

PQN

-RRB-

and

the

Ylm

-LRB-

θ

ϕ

-RRB-

are

the

normalized

real

spherical

harmonics

The

PQN

are

those

of

the

valence

shell

i.e.

the

set

of

atomic

orbitals

most

important

in

forming

chemical

bonds

For

the

PQN

used

are

shown

in

Table

For

most

main-group

elements

the

s

and

p

PQN

are

the

same

and

when

d

orbitals

are

present

all

three

PQN

are

the

same

that

is

the

PQN

are

-LRB-

ns

np

nd

-RRB-

For

transition

metals

the

d

PQN

is

one

less

than

that

of

the

s

and

p

shells

i.e.

-LRB-

ns

np

-LRB-

n

-RRB-

d

-RRB-

An

exception

to

this

generalization

occurs

in

the

elements

of

Group

VIII

Here

the

valence

shell

is

completely

filled

so

in

all

chemical

interactions

that

could

occur

between

an

atom

of

a

Group

VIII

element

and

any

other

atom

electron

density

could

only

migrate

from

the

Group

VIII

element

to

the

other

atom

That

is

when

a

rare

gas

element

forms

any

type

of

chemical

bond

it

would

necessarily

become

slightly

positive

This

is

an

unrealistic

result

In

order

to

allow

rare

gas

atoms

to

have

the

potential

of

being

slightly

negative

the

set

of

valence

orbitals

was

changed

from

-LRB-

ns

np

-RRB-

to

-LRB-

np

-LRB-

n

-RRB-

s

-RRB-

for

the

elements

Ne

Ar

Kr

and

Xe

Helium

is

the

only

exception

to

this

change

because

it

does

not

have

a

valence

shell

For

helium

the

valence

shell

used

was

-LRB-

-RRB-

this

being

considered

the

best

compromise

Table

quantum

numbers

for

atomic

orbitals

spd

Parameter

optimization

Background

The

objective

of

parameter

optimization

is

to

modify

the

values

of

the

parameters

so

as

to

minimize

the

error

function

S

Eq

-LRB-

-RRB-

representing

the

square

of

the

differences

between

the

values

of

reference

data

Qref

-LRB-

i

-RRB-

and

the

values

calculated

using

the

semiempirical

method

Qcalc

-LRB-

i

-RRB-

with

appropriate

weighting

factors

gi

This

process

is

initiated

by

rendering

the

reference

data

in

the

training

set

dimensionless

The

default

conversion

factors

are

given

in

Table

with

weighting

factors

for

reference

data

represented

by

rules

being

much

larger

typically

in

the

order

of

kcal

mol

Table

weighting

factors

for

reference

dataReference

mol.kcal

Å

degrees

Debye

P.

eV

Main

Transition

The

elements

were

divided

into

four

sets

core

elements

-LRB-

H

C

N

and

O

-RRB-

other

elements

important

in

organic

chemistry

-LRB-

F

P

S

Cl

K

Br

I

-RRB-

the

rest

of

the

main

group

and

the

transitions

metals

Elements

were

assigned

to

the

different

sets

based

on

their

presumed

degree

of

importance

in

biochemistry

and

this

importance

was

converted

into

a

weighting

factor

to

be

used

in

the

parameterization

optimization

procedure

Reference

data

representing

species

consisting

only

of

core

elements

were

given

their

default

weight

When

other

elements

were

present

the

weight

was

set

to

the

default

weight

times

the

smallest

multiplier

shown

in

Table

Thus

the

default

weight

for

a

reference

datum

involving

tetramethyllead

Pb

-LRB-

-RRB-

would

be

multiplied

by

reflecting

the

fact

that

this

species

contains

an

element

in

the

main

group

set

For

a

given

set

of

parameters

P

optimization

proceeds

by

calculating

the

values

of

all

the

Qcalc

-LRB-

i

-RRB-

their

first

derivatives

with

respect

to

each

parameter

P

-LRB-

j

-RRB-

and

the

second

derivatives

with

respect

to

every

pair

of

parameters

Evaluating

these

quantities

is

time-consuming

and

considerable

effort

was

expended

in

minimizing

the

need

for

explicit

evaluation

of

these

functions

The

most

efficient

strategy

developed

-LSB-

-RSB-

involved

assuming

that

in

the

region

of

parameter

space

near

to

the

current

values

of

the

parameters

the

values

of

the

first

derivatives

of

the

Qcalc

-LRB-

i

-RRB-

with

respect

to

P

were

at

least

to

a

first

approximation

constant

By

making

this

assumption

the

values

of

the

parameters

could

then

be

updated

using

perturbation

theory

Because

the

assumption

is

only

valid

in

the

region

of

the

starting

point

in

parameter

space

periodically

the

focus

was

moved

to

the

new

point

in

parameter

space

and

a

complete

explicit

re-evaluations

of

all

the

functions

performed

The

parameter

optimization

process

terminated

when

the

scalar

of

the

first

derivatives

dropped

below

a

preset

limit

This

process

was

fully

automated

and

for

given

sets

of

reference

data

and

parameters

parameter

optimization

could

be

performed

rapidly

easily

and

reliably

Sequence

of

optimization

of

parameters

Notwithstanding

the

reliability

of

the

parameter

optimization

procedure

a

simple

global

optimization

of

all

the

parameters

for

all

elements

involving

about

over

discrete

species

was

found

to

be

impractical

because

of

the

large

number

of

derivatives

involved

Such

an

optimization

would

involve

over

parameters

and

over

reference

data

The

set

of

second

derivatives

alone

would

consist

of

terms

With

more

powerful

computers

evaluating

such

large

sets

of

derivatives

might

be

practical

some

day

but

even

then

one

faulty

reference

datum

or

one

faulty

initial

parameter

value

would

ruin

an

optimization

run

The

strategy

of

parameter

optimization

was

approached

with

great

caution

and

the

procedure

finally

adopted

was

as

follows

Because

the

elements

H

C

N

and

O

are

of

paramount

importance

in

biochemistry

and

because

large

amounts

of

reference

data

are

available

the

starting

point

for

parameter

optimization

involved

the

simultaneous

optimization

of

parameters

for

these

four

elements

For

the

purposes

of

discussion

this

set

of

four

elements

will

be

called

the

core

elements

Once

stable

parameters

had

been

obtained

parameters

for

other

elements

important

in

organic

chemistry

were

optimized

in

two

stages

First

the

parameters

for

the

core

elements

were

held

constant

and

parameters

for

the

elements

F

P

S

Cl

Br

and

I

were

optimized

one

at

a

time

Then

all

parameters

for

all

ten

elements

were

simultaneously

optimized

This

set

-LRB-

the

organic

elements

-RRB-

was

then

used

as

the

starting

point

for

parameterizing

the

rest

of

the

main

group

The

same

sequence

was

followed

for

the

rest

of

the

main-group

elements

That

is

parameters

for

each

element

were

optimized

while

freezing

the

parameters

for

the

organic

elements

Then

once

all

the

elements

had

been

processed

all

parameters

for

all

of

the

main-group

elements

plus

zinc

cadmium

and

mercury

were

optimized

simultaneously

When

parameters

for

the

transition

metals

were

being

optimized

all

parameters

for

the

main

group

elements

were

held

constant

There

were

several

reasons

for

this

Most

importantly

the

reference

data

for

the

transition

metals

particularly

the

thermochemical

data

was

of

lower

quality

so

one

consideration

was

to

prevent

the

transition

metals

from

having

a

deleterious

effect

on

the

main-group

elements

Another

important

consideration

was

that

most

compounds

involving

transition

metals

also

involved

only

elements

of

the

organic

set

Since

parameters

for

these

elements

had

been

optimized

using

a

training

set

consisting

of

all

the

main-group

elements

the

values

of

the

optimized

parameters

would

likely

be

relatively

insensitive

to

the

influence

of

the

small

number

of

additional

reference

data

involving

transition

metals

In

general

all

parameters

for

a

given

element

were

optimized

simultaneously

this

was

both

efficient

and

convenient

In

some

optimizations

specifically

those

involving

a

new

element

only

sub-sets

of

parameters

were

used

Three

main

sub-sets

were

used

Parameters

that

determine

atomic

electronic

properties

For

most

elements

atomic

energy

levels

are

determined

by

six

parameters

the

one-electron

one-center

integrals

Uss

Upp

Udd

and

the

internal

orbital

exponents

ζsn

ζpn

and

ζdn

If

the

heat

of

ionization

and

sufficient

atomic

energy

level

data

were

available

these

quantities

could

be

uniquely

defined

there

would

be

no

need

for

the

use

of

molecular

reference

data

These

parameters

were

the

first

to

be

optimized

whenever

an

optimization

was

started

for

an

element

that

had

not

previously

been

parameterized

Parameters

that

determine

molecular

electronic

properties

Two

of

the

more

important

electronic

molecular

properties

are

the

dipole

moment

which

indicates

the

degree

of

polarization

within

a

molecule

and

the

ionization

potential

These

properties

are

determined

primarily

by

parameters

the

six

parameters

that

determine

atomic

electronic

properties

and

six

additional

parameters

βs

βp

and

βd

and

the

Slater

orbital

exponents

ζs

ζp

and

ζd

In

the

second

stage

of

parameter

optimization

the

first

six

parameters

were

held

constant

at

the

values

defined

using

atomic

data

and

the

second

set

optimized

During

this

operation

all

geometries

were

fixed

at

their

reference

values

Parameters

that

determine

geometries

As

soon

as

an

initial

optimized

set

of

electronic

parameters

was

available

the

diatomic

and

other

core-core

parameters

could

be

optimized

The

most

efficient

process

was

to

optimize

these

parameters

initially

without

allowing

the

electronic

parameters

or

the

molecular

geometries

to

optimize

If

geometries

were

allowed

to

optimize

optimization

of

the

core-core

parameters

would

be

slowed

considerably

because

of

the

tight

dependency

of

the

optimized

geometries

on

the

values

of

the

core-core

parameters

and

vice

versa

As

soon

as

all

parameters

had

been

optimized

using

fixed

geometries

the

geometries

were

allowed

to

relax

and

the

parameters

that

determine

geometry

re-optimized

After

that

there

would

be

three

sets

of

incompletely

optimized

parameters

the

six

atomic

electronic

parameters

the

six

molecular

electronic

parameters

and

the

core-core

parameters

The

only

remaining

operation

was

the

simultaneous

optimization

of

all

the

parameters

If

the

training

set

of

reference

data

was

insufficient

to

unambiguously

define

the

values

of

all

the

parameters

then

at

that

stage

the

potential

existed

for

the

parameters

to

become

ill-defined

An

example

of

this

would

be

where

there

were

too

few

atomic

energy

levels

to

allow

all

six

parameters

in

the

first

set

to

be

defined

To

allow

for

this

a

penalty

function

was

added

to

each

parameter

If

the

values

of

a

parameter

exceeded

pre-defined

limits

the

error

function

S

was

incremented

by

a

constant

times

the

square

of

the

excess

No

penalty

was

applied

if

the

value

of

a

parameter

was

between

the

pre-defined

limits

that

is

no

bias

was

applied

to

the

numerical

value

of

a

parameter

During

the

early

stages

of

simultaneous

optimization

of

all

the

parameters

for

a

given

element

the

penalty

function

was

used

frequently

In

the

later

stages

the

penalty

function

was

invoked

rarely

and

then

only

when

there

was

a

distinct

shortage

of

reference

data

Results

Parameters

for

atomic

and

diatomic

parameters

for

the

elements

are

presented

in

Tables

and

respectively

Not

all

elements

have

all

parameters

where

monatomic

parameters

are

missing

the

associated

approximations

were

not

used

For

diatomic

parameters

where

an

atom-pair

is

missing

no

representatives

of

that

type

of

bond

were

used

Table

parameters

for

elementsTable

core

core

parameters

Accuracy

Comparison

with

other

semiempirical

methods

Using

the

program

-LSB-

-RSB-

an

extensive

comparison

was

made

between

the

results

obtained

using

and

those

from

and

This

comparison

was

started

by

generating

tables

of

reference

data

-LRB-

that

is

ΔHf

geometries

ionization

potentials

-LRB-

I.P.s

-RRB-

and

dipole

moments

-RRB-

and

differences

between

the

calculated

and

reference

values

using

each

of

the

four

methods

presented

here

Because

of

their

size

they

are

provided

in

the

supplementary

material

To

simplify

navigating

within

the

tables

all

species

are

listed

in

the

order

of

their

empirical

formula

Average

unsigned

errors

-LRB-

AUE

-RRB-

for

ΔHf

for

each

element

parameterized

at

the

level

are

shown

in

Table

together

with

AUE

for

and

The

number

of

data

used

in

each

average

varies

depending

on

the

elements

available

in

each

method

boron

-LSB-

-RSB-

uses

a

different

core-core

interaction

expression

from

the

other

elements

and

was

not

used

AUE

for

bond-lengths

are

shown

in

Table

In

those

cases

where

a

calculated

bond-length

was

very

large

indicating

that

the

bond

had

broken

the

bond-length

was

not

used

in

the

analysis

If

such

data

had

been

used

the

resulting

statistics

would

have

been

misleading

AUE

for

angles

are

shown

in

Table

Errors

in

angles

for

many

elements

that

form

very

ionic

i.e.

labile

bonds

are

of

less

importance

than

errors

involving

elements

that

form

strong

covalent

bonds

The

angles

subtended

by

such

bonds

are

often

determined

largely

by

the

electronic

structure

of

the

atom

Information

on

the

accuracy

of

prediction

of

molecular

electronic

structure

can

also

be

inferred

from

the

AUE

of

dipole

moments

Table

and

ionization

potentials

Table

Table

unsigned

errors

in

calculated

heats

of

formation

-LRB-

kcal

mol

-RRB-

ElementPM6No.PM5No.PM3No.AM1No.Hydrogen7

Table

unsigned

errors

in

bond

lengths

-LRB-

Å

-RRB-

ElementPM6No.PM5No.PM3No.AM1No.Hydrogen0

Table

unsigned

errors

in

bond

angles

-LRB-

Degrees

-RRB-

in

in

in

in

Table

unsigned

errors

in

dipole

moments

-LRB-

D

-RRB-

Table

unsigned

errors

in

ionization

potential

-LRB-

eV

-RRB-

Comparison

of

the

accuracy

of

with

the

other

NDDO

methods

and

was

made

more

complicated

by

the

fact

that

different

sets

of

elements

were

available

in

each

method

To

allow

a

simple

comparison

therefore

average

unsigned

errors

-LRB-

AUE

-RRB-

for

the

four

common

properties

for

various

subsets

are

presented

in

Tables

and

To

ensure

a

valid

comparison

the

same

number

of

data

were

used

in

each

method

except

for

in

whole

of

main

group

where

data

for

cadmium

and

boron

were

not

used

Table

unsigned

errors

in

ΔHf

for

various

sets

of

elements

-LRB-

kcal

mol

-RRB-

Set

of

C

N

H

C

N

O

F

P

S

Cl

Br

I

Whole

of

main

Table

unsigned

errors

in

bond

lengths

for

various

sets

of

elements

-LRB-

Å

-RRB-

Set

of

C

N

H

C

N

O

F

P

S

Cl

Br

Whole

of

main

Table

unsigned

errors

in

angles

for

various

sets

of

elements

-LRB-

Degrees

-RRB-

Set

of

C

N

H

C

N

O

F

P

S

Cl

Br

Whole

of

main

Table

unsigned

errors

in

dipole

moments

for

various

sets

of

elements

-LRB-

D

-RRB-

Set

of

C

N

H

C

N

O

F

P

S

Cl

Br

Whole

of

main

Table

unsigned

errors

in

I.P.s

for

various

sets

of

elements

-LRB-

eV

-RRB-

Set

of

C

N

H

C

N

O

F

P

S

Cl

Br

Whole

of

main

Comparison

with

Winget

et

al.

-LSB-

-RSB-

developed

parameters

for

P

S

and

Cl

in

which

Voityuk

's

diatomic

parameters

were

used

for

all

atom-pairs

involving

P

S

and

Cl

with

H

C

N

O

F

P

S

Cl

and

Mo.

In

the

method

all

parameters

for

elements

other

than

the

ones

being

optimized

are

held

constant

at

the

values

As

such

could

be

regarded

as

a

hybrid

method

parameters

for

a

few

individual

elements

are

re-optimized

in

this

case

with

some

changes

in

the

set

of

approximations

while

holding

the

parameters

for

the

other

methods

constant

at

their

values

Tables

comparing

individual

P

S

and

Cl

species

calculated

with

and

are

given

in

the

supplementary

material

A

summary

of

the

statistical

analysis

is

given

in

Table

Winget

et

al.

also

reported

parameters

for

titanium

and

zirconium

-LSB-

-RSB-

These

parameters

were

not

used

in

the

comparison

given

here

because

the

set

of

approximations

used

was

incompatible

with

the

set

used

in

Table

unsigned

errors

in

phosphorus

sulfur

and

chlorine

ΔHf

-LRB-

kcal

mol

-RRB-

Bond

length

-LRB-

Å

-RRB-

Dipole

-LRB-

D

-RRB-

I.P.

-LRB-

eV

-RRB-

Angles

-LRB-

Degrees

-RRB-

.319.1900.0220.051560.570.49100.510.81122.53.319

.510.61990.0290.060710.360.64140.520.50453.14.134

.118.21560.0250.106690.550.60100.520.62253.414.64

Comparison

with

In

ten

elements

H

C

N

O

F

P

S

Cl

Br

and

I

that

had

been

parameterized

at

the

level

were

re-parameterized

-LSB-

-RSB-

the

result

was

a

new

method

No

changes

were

made

to

the

set

of

approximations

used

so

that

for

example

P

S

Cl

Br

and

I

used

only

the

s-p

basis

set

That

is

was

functionally

identical

to

A

statistical

analysis

showed

that

was

more

accurate

than

any

of

the

other

NDDO

methods

and

therefore

was

the

method

of

choice

for

modeling

organic

compounds

An

indication

of

the

effect

of

the

current

changes

to

the

set

of

approximations

can

be

obtained

by

comparing

the

AUE

for

and

in

Tables

and

Voityuk

reported

the

parameterization

of

molybdenum

-LSB-

-RSB-

at

the

level

These

parameters

were

added

to

the

standard

parameters

and

were

used

in

the

analysis

Comparison

with

high-level

methods

A

comparison

of

HF

-LRB-

d

-RRB-

and

-LRB-

d

-RRB-

errors

in

predicted

ΔHf

for

compounds

is

given

in

the

supplementary

material

Only

compounds

containing

the

elements

H

C

N

O

F

P

S

Cl

and

Br

were

considered

these

being

the

more

important

elements

in

biochemistry

Ab-initio

ΔHf

were

obtained

from

the

calculated

total

energies

by

the

addition

of

a

simple

atomic

correction

and

conversion

from

atomic

units

to

kcal

mol

No

allowance

was

made

for

thermal

population

effects

zero

point

energies

etc.

the

assumption

being

made

that

such

effects

could

be

absorbed

into

the

atomic

corrections

A

statistical

analysis

of

errors

in

thermochemical

predictions

for

the

three

methods

is

given

in

Table

A

check

was

also

done

to

verify

that

the

error

distribution

was

approximately

Gaussian

The

resulting

histogram

shown

in

Fig.

shows

that

the

distribution

is

indeed

Gaussian

Table

analysis

of

errors

in

predicted

ΔHf

for

various

methods

-LRB-

kcal

mol

-RRB-

HF

No

of

molecules

in

set

Basis

set

Fig.

of

errors

in

calculated

ΔHf

Hydrogen

bonding

One

of

the

commonest

forms

of

hydrogen

bonding

involves

a

hydrogen

atom

attached

to

an

oxygen

atom

and

forming

a

weak

bond

to

a

distant

oxygen

atom

The

simplest

well-characterized

case

is

that

of

the

water

dimer

In

an

exhaustive

analysis

of

this

system

Tschumper

et

al.

-LSB-

-RSB-

characterized

this

system

using

CCSD

-LRB-

T

-RRB-

and

a

large

basis

set

They

identified

and

characterized

ten

stationary

points

on

the

potential

energy

surface

of

the

dimer

and

determined

that

the

lowest

energy

conformer

of

the

water

dimer

was

kcal

mol

more

stable

than

two

isolated

water

molecules

A

comparison

of

the

relative

heats

of

formation

of

these

points

calculated

using

NDDO

methods

is

shown

in

Table

The

AUE

for

the

various

methods

are

as

follows

kcal

mol

and

Table

energies

of

conformers

of

water

dimerStructureRef.Relative

ΔHf

-LRB-

kcal

mol

-RRB-

-LRB-

Non-planar

open

Cs

-RRB-

-LRB-

Non

planar

open

Cs

-RRB-

-LRB-

Open

Ci

-RRB-

-LRB-

Planar

Open

Cs

-RRB-

-LRB-

Cyclic

Ci

-RRB-

-LRB-

Cyclic

-RRB-

-LRB-

Cyclic

-RRB-

-LRB-

Triply

Hydrogen

-LRB-

Non-planar

-LRB-

Non-planar

-LRB-

Planar

Bifurcated

Relative

to

two

isolated

water

molecules

Structures

are

relative

to

Structure

The

energies

of

various

different

types

of

hydrogen

bonds

were

estimated

from

the

energy

released

when

the

two

small

molecules

involved

associate

to

form

a

hydrogen-bonded

system

Table

lists

the

values

predicted

using

and

the

NDDO

methods

Table

of

and

hydrogen

bond

energies

-LRB-

kcal

mol

-RRB-

Hydrogen-bonded

ammonia

Water

methanol

Water

acetone

Water

dimer

linear

-LRB-

O

H

O

-RRB-

Water

dimer

Benzene

dimer

T-shaped

Water

acetate

anion

Water

formaldehyde

Water

ammonia

Water

formamide

Formic

acid

dimer

Water

methylammonium

cation

Formamide

formamide

Acetic

acid

dimer

Nitrogen

pyramidalization

A

well-documented

fault

in

nitrogen

was

its

exaggerated

degree

of

pyramidalization

when

in

the

configuration

This

is

dramatically

evident

in

N-methylacetamide

where

the

H-N-C

C

torsion

angle

should

be

but

is

predicted

by

to

be

That

is

the

nitrogen

instead

of

being

in

a

planar

environment

is

predicted

to

be

highly

pyramidal

The

results

of

a

survey

of

molecules

that

contain

nitrogen

are

presented

in

Table

Table

errors

in

pyramidalization

of

nitrogen

-LRB-

Torsion

angle

about

nitrogen

in

degrees

-RRB-

signed

error

Average

unsigned

Transition

metals

Optimizing

parameters

for

transition

metals

was

not

as

straightforward

as

for

the

main

group

elements

As

with

the

main

group

compounds

there

is

a

wealth

of

structural

reference

data

on

transition

metal

complexes

However

unlike

main

group

compounds

there

is

a

distinct

shortage

of

reliable

thermochemical

data

To

alleviate

this

shortage

the

thermochemical

data

that

was

available

was

augmented

by

the

results

of

DFT

calculations

It

was

recognized

however

that

these

derived

reference

data

were

likely

to

be

of

a

lower

accuracy

than

the

experimental

data

Many

transition

metal

complexes

are

also

highly

labile

a

consequence

of

this

was

that

some

moieties

that

are

known

to

exist

in

the

solid

phase

were

predicted

to

be

unstable

in

the

gas

phase

at

least

at

the

level

of

calculation

In

most

cases

such

moieties

had

a

high

formal

charge

therefore

without

any

countercharge

their

instability

in

isolation

is

understandable

When

an

intrinsically

unstable

ion

was

identified

it

was

removed

from

further

consideration

Most

transition

metal

compounds

also

have

extensive

UV-visible

properties

arising

from

d-d

transitions

and

from

charge-transfer

excitations

the

presence

of

these

absorption

bands

being

indicative

of

the

existence

of

low-lying

electronic

excited

states

The

self-consistent

field

-LRB-

SCF

-RRB-

equations

frequently

did

not

converge

unless

special

techniques

were

used

One

of

these

using

the

direct

inversion

of

the

iterative

sub-space

-LSB-

-RSB-

or

DIIS

would

frequently

yield

an

SCF

when

other

methods

failed

However

as

a

result

of

the

way

it

works

the

DIIS

converged

the

wavefunction

to

the

nearest

stationary

point

not

necessarily

to

the

lowest

energy

point

Because

of

the

potential

existence

of

multiple

low-lying

excited

states

special

care

had

to

be

taken

when

the

DIIS

technique

was

used

Conversely

the

tendency

to

converge

to

the

nearest

stationary

point

was

an

advantage

when

electronic

states

of

transition

metal

atoms

were

being

optimized

In

several

instances

the

lowest

energy

wavefunction

corresponded

to

a

hybrid

of

s

p

and

d

atomic

orbitals

that

did

not

transform

as

any

irreducible

representation

of

the

group

of

the

sphere

In

those

cases

the

wavefunction

could

be

induced

to

converge

to

the

correct

spherical

harmonic

solution

by

using

the

DIIS

procedure

Sets

of

transition

metals

For

the

purpose

of

discussion

the

set

of

transition

metals

can

be

partitioned

into

eight

of

the

groups

of

the

Periodic

Table

with

each

group

containing

one

or

more

triads

of

elements

A

detailed

discussion

of

each

element

is

impractical

because

of

the

wide

range

of

compounds

in

transition

metal

chemistry

The

following

section

therefore

will

be

limited

to

systems

where

does

not

work

well

and

to

systems

illustrative

of

the

structural

chemistry

of

specific

elements

Group

IIIA

Scandium

Yttrium

Lanthanum

and

Lutetium

Possibly

because

of

its

scarcity

only

a

few

experimental

thermochemical

reference

data

for

scandium

compounds

were

available

for

use

in

the

parameterization

What

reference

data

existed

were

augmented

by

the

results

of

DFT

calculations

and

with

a

large

number

of

atomic

energy

levels

for

the

neutral

and

ionized

atom

Only

the

chemistry

of

ScIII

was

studied

Most

bond

lengths

involving

scandium

were

reproduced

with

good

accuracy

-LRB-

for

example

tri

-LRB-

η5

cyclopentadienyl

-RRB-

scandium

Fig.

-RRB-

the

exception

being

the

coordination

complex

-LSB-

Sc

-LRB-

-RRB-

-RSB-

which

predicts

to

decompose

to

-LSB-

Sc

-LRB-

-RRB-

-RSB-

plus

two

water

molecules

Fig.

-LRB-

η5

cyclopentadienyl

-RRB-

scandium

Reference

value

in

parenthesis

As

with

scandium

very

few

thermochemical

reference

data

were

found

for

yttrium

or

lanthanum

To

compensate

for

this

extensive

use

was

made

of

the

CSD

The

chemistry

of

lutetium

is

similar

to

that

of

lanthanum

with

the

principal

difference

being

that

whereas

LaIII

has

an

empty

shell

in

LuIII

that

shell

is

completely

filled

Since

the

shell

is

at

least

chemically

virtually

inert

lutetium

could

be

regarded

as

a

conventional

transition

metal

and

was

therefore

included

in

this

work

Group

IVA

Titanium

Zirconium

and

Hafnium

In

contrast

to

all

the

elements

of

Group

IIIA

titanium

is

plentiful

and

an

abundance

of

reference

data

on

TiIII

and

the

more

common

TiIV

is

available

These

data

include

many

tetrahedral

and

octahedral

inorganic

complexes

as

well

as

organotitanium

compounds

Most

bond

lengths

are

reproduced

with

good

accuracy

the

exceptions

being

the

Ti-H

bond

in

where

the

predicted

value

Å

is

Å

shorter

than

the

reference

and

coordination

complexes

which

involve

oxygen

forming

a

purely

dative

bond

to

titanium

In

this

latter

case

the

Ti-O

bond

is

typically

too

long

by

to

Å

The

behavior

of

zirconium

and

hafnium

is

similar

to

that

of

titanium

Group

VA

Vanadium

Niobium

and

Tantalum

Most

of

the

structural

chemistry

of

vanadium

in

its

five

common

oxidation

states

II

III

IV

and

V

are

reproduced

with

good

accuracy

The

common

structure

which

occurs

in

bis

-LRB-

Acetylacetonato

-RRB-

oxo-vanadium

-LRB-

iv

-RRB-

where

vanadium

forms

a

double

bond

to

one

oxygen

atom

and

single

bonds

to

the

other

four

is

reproduced

accurately

the

V

=O

distance

being

Å

-LRB-

reference

-RRB-

the

V-O

distance

Å

-LRB-

-RRB-

and

the

O-V

=O

angle

-LRB-

-RRB-

Not

all

systems

were

reproduced

with

such

accuracy

When

there

are

several

ligands

around

a

vanadium

atom

the

effects

of

steric

crowding

are

over-emphasized

and

incorrectly

predicts

that

one

of

the

metal-oxygen

bonds

would

break

An

example

is

bis

-LRB-

bis

-LRB-

O

-RRB-

-LRB-

-RRB-

vanadium

-RRB-

where

each

vanadium

atom

extends

bonds

to

four

oxygen

atoms

and

one

cyclopentadienyl

In

this

system

predicts

that

one

of

the

V-O

bonds

would

break

In

the

heavier

elements

there

is

an

increased

tendency

to

form

highly

symmetric

polynuclear

complexes

An

example

is

the

tantalum

dication

-LSB-

-RSB-

This

is

predicted

to

have

an

octahedral

structure

in

modest

agreement

with

the

DFT

result

-LRB-

Fig.

-RRB-

Fig.

structure

of

the

complex

ion

-LSB-

-RSB-

Reference

value

in

parenthesis

Transition

metal

complexes

usually

have

one

or

more

unpaired

electrons

such

systems

can

only

be

modeled

using

an

open

shell

method

such

as

unrestricted

Hartree

Fock

-LRB-

UHF

-RRB-

or

restricted

Hartree

Fock

followed

by

a

configuration

interaction

-LRB-

RHF-CI

-RRB-

correction

The

UHF

method

is

faster

and

more

reliable

and

is

the

method

of

choice

when

only

simple

properties

such

as

heats

of

formation

or

geometries

are

of

interest

For

-LSB-

-RSB-

M

Nb

or

Ta

X

Cl

or

Br

UHF

predicts

an

almost

octahedral

complex

a

very

slight

distortion

lowering

the

symmetry

to

This

distortion

is

also

reflected

in

the

asymmetric

charge

distribution

When

RHF-CI

is

used

the

geometry

converges

on

the

exact

Oh

structure

Group

VIA

Chromium

Molybdenum

and

Tungsten

Most

Cr

O

and

Cr

N

bonds

are

reproduced

well

as

illustrated

by

-LSB-

CrIII

-LRB-

EDTA

-RRB-

-RSB-

in

Fig.

The

organometallic

bond

in

chromium

hexacarbonyl

is

Å

which

is

in

good

agreement

with

the

crystal

structure

Å

found

in

-LSB-

-RSB-

Fig.

Ethylenediaminetetraacetate

anion

-LSB-

Cr

-LRB-

III

-RRB-

-LRB-

EDTA

-RRB-

-RSB-

The

octacyano-molybdate

-LRB-

IV

-RRB-

moiety

-LSB-

MoIV

-LRB-

CN

-RRB-

-RSB-

is

a

stable

eight-coordinate

organometallic

molybdenum

complex

ion

whose

geometry

in

the

crystal

is

that

of

a

slightly

distorted

square

antiprism

Rather

unexpectedly

this

structure

was

reproduced

by

the

expectation

being

that

in

the

absence

of

crystal

field

forces

the

structure

would

have

optimized

to

a

geometry

which

has

a

higher

symmetry

i.e.

converged

to

the

exact

geometry

The

predicted

Mo-C

distance

was

versus

Å

again

in

unexpectedly

good

agreement

for

an

ion

with

such

a

large

formal

charge

Molybdenum

forms

the

cluster

anion

-LSB-

-LRB-

-RRB-

-RSB-

in

which

the

six

molybdenum

atoms

form

a

regular

octahedron

successfully

reproduces

this

structure

and

predicts

the

following

distances

Mo-Mo

-LRB-

-RRB-

-LRB-

-RRB-

and

Mo-Cl

-LRB-

Å

-RRB-

The

trioxide

of

molybdenum

can

form

polyoxometalates

a

typical

example

of

which

is

the

α-keggin

heteropolyoxyanion

-LSB-

-RSB-

In

this

structure

shown

in

Fig.

each

Mo

forms

a

double

bond

with

one

oxygen

single

bonds

to

four

other

oxygen

atoms

and

what

can

only

be

described

as

a

third

of

a

bond

to

a

sixth

oxygen

that

is

part

of

the

unit

Despite

the

apparently

high

symmetry

Td

this

system

has

only

a

center

of

inversion

This

low

symmetry

is

reproduced

by

Fig.

structure

of

tetraconta-oxo-silicon-dodeca-molybdenum

-LSB-

-RSB-

Crossed-eyes

stereo

Mo

=O

Å

-LRB-

-RRB-

Mo-O

-LRB-

-RRB-

Si

O

-LRB-

-RRB-

-LRB-

Ref

in

parentheses

-RRB-

predicts

the

structures

of

all

three

hexacarbonyls

with

good

accuracy

but

gives

qualitatively

the

wrong

structures

for

the

dinuclear

decacarbonyls

This

failure

to

qualitatively

predict

the

structure

of

the

polynuclear

carbonyls

occurred

frequently

during

the

survey

of

the

transition

metals

Group

VIIA

Manganese

Technetium

and

Rhenium

Like

many

other

transition

metals

manganese

can

form

sepulchrates

closo

polyhedral

complexes

of

general

structure

-LRB-

-RRB-

icosane

In

contrast

to

the

more

common

open

hexadentate

chelates

of

manganese

e.g.

-LSB-

MnII

-LRB-

EDTA

-RRB-

-RSB-

the

metal

atom

in

a

sepulchrate

is

extremely

tightly

bound

and

can

not

be

removed

without

destroying

the

organic

framework

A

simple

sepulchrate

is

shown

in

Fig.

predicts

the

Mn-N

distance

with

good

accuracy

but

gets

the

twist

angle

incorrect

A

DFT

calculation

reproduced

the

twist

angle

found

in

the

crystal

which

suggests

that

the

error

in

the

twist

angle

can

not

be

attributed

to

the

neglect

of

crystal

packing

forces

Fig.

-LSB-

Sepulchrate-manganese

-LRB-

III

-RRB-

-RSB-

-LRB-

-LRB-

-RRB-

icosane

-RRB-

manganese

-LRB-

III

-RRB-

CSD

entry

HAFBUL

Although

there

is

a

large

amount

of

structural

information

on

technetium

compounds

there

is

a

distinct

shortage

of

thermochemical

data

To

make

up

for

this

almost

all

the

reference

heats

of

formation

of

representative

technetium

compounds

were

derived

from

DFT

calculations

Only

one

heat

of

formation

was

used

in

this

derivation

that

of

the

isolated

technetium

atom

therefore

the

reference

values

used

almost

certainly

include

a

systematic

error

that

may

amount

to

many

kilocalories

per

mole

Consequently

the

reference

heats

of

formation

and

the

errors

in

predicted

heats

of

formation

of

technetium

compounds

should

be

taken

cum

granus

salis

However

this

should

not

be

construed

as

implying

that

they

are

meaningless

because

reactions

are

balanced

when

heats

of

reaction

are

evaluated

any

systematic

errors

in

the

heats

of

formation

are

cancelled

out

One

of

the

more

important

technetium

species

is

the

pertechnetate

ion

-LSB-

-RSB-

used

in

nuclear

medicine

In

this

ion

predicts

the

Tc-O

distance

to

be

Å

in

good

agreement

with

the

DFT

value

of

Å

Group

VIIIA

Iron

Cobalt

Nickel

Ruthenium

Rhodium

Palladium

Osmium

Iridium

and

Platinum

The

geometries

of

most

compounds

of

this

large

group

were

reproduced

with

modest

to

good

accuracy

including

the

iron-porphorin

complex

Fig.

of

the

type

found

in

heme

The

main

exception

is

iron

pentacarbonyl

Fe

-LRB-

CO

-RRB-

which

in

its

equilibrium

geometry

is

known

unambiguously

to

be

of

point-group

and

which

predicts

to

be

equally

unambiguously

When

this

error

was

discovered

attempts

were

made

to

correct

the

fault

by

adding

a

rule

to

the

training

set

for

iron

This

rule

stated

that

The

geometry

was

kcal

mol

higher

in

energy

than

the

geometry

kcal

mol

being

the

difference

between

the

energies

of

the

two

structures

calculated

using

DFT

However

even

when

a

very

large

weighting

factor

was

used

the

structure

remained

more

stable

than

the

albeit

the

error

in

the

relative

energies

was

decreased

During

this

optimization

errors

in

all

other

iron

compounds

increased

significantly

Rather

than

accept

a

general

deterioration

in

the

predicted

properties

of

iron

compounds

the

rule

was

removed

from

the

training

set

Fig.

7trans-7,8-Dihydro-2,3,7,8,12,13,17,18-octaethylporphyrinato-iron

-LRB-

II

-RRB-

Reference

value

-LRB-

CSD

entry

BUYKUB

-RRB-

in

parenthesis

The

well-known

red

complex

nickel

dimethylglyoxime

is

normally

encountered

in

the

quantitative

analysis

of

inorganic

nickel

in

solution

At

the

center

of

the

molecule

is

the

planar

structure

structure

which

is

frequently

found

in

nickel

compounds

in

biochemical

systems

predicts

this

with

good

accuracy

-LRB-

Fig.

-RRB-

Fig.

Dimethylglyoxime

Reference

value

-LRB-

CSD

entry

-RRB-

in

parenthesis

One

of

the

first

polyhapto

organometallic

complexes

discovered

was

Zeise

's

salt

In

the

anion

-LSB-

-LRB-

-RRB-

-RSB-

platinum

forms

a

synergic

bond

with

an

ethylene

molecule

The

calculated

and

X-ray

structures

of

this

complex

are

shown

in

Fig.

Fig.

's

Salt

trichloro

-LRB-

-RRB-

platinate

Reference

value

-LRB-

CSD

entry

XIVSAK

-RRB-

in

parenthesis

Group

IB

Copper

Silver

and

Gold

Copper

phthalocyanine

is

an

extremely

stable

blue

dyestuff

As

with

nickel

dimethylglyoxime

the

planar

moiety

at

the

center

of

the

porphyrin

ring

is

typical

of

many

copper

species

of

importance

in

biochemistry

reproduces

it

with

very

good

accuracy

-LRB-

Fig.

-RRB-

Fig.

phthalocyanine

Reference

value

-LRB-

CSD

entry

-RRB-

in

parenthesis

Dimethyl

gold

cyanide

tetramer

provides

a

good

example

of

a

square-planar

AuIII

complex

In

this

system

each

gold

atom

forms

covalent

single

bonds

of

length

Å

-LRB-

-RRB-

to

the

carbons

of

the

methyl

groups

a

weaker

longer

bond

of

length

Å

-LRB-

-RRB-

to

the

carbon

of

the

cyanide

group

and

a

still

longer

bond

Å

-LRB-

-RRB-

to

the

nitrogen

atom

Gold

also

forms

small

planar

clusters

predicts

that

neutral

clusters

of

up

to

about

nine

gold

atoms

should

be

planar

an

example

being

the

cluster

in

which

the

Au-Au

distance

is

predicted

to

be

Å

-LRB-

-RRB-

Clusters

of

up

to

gold

atoms

are

also

predicted

to

be

stable

provided

the

cluster

has

a

single

negative

charge

Group

IIB

Zinc

Cadmium

and

Mercury

These

elements

have

completely

filled

d

shells

therefore

the

valence

shell

can

be

limited

to

the

s

and

p

orbitals

As

such

they

behave

like

main-group

elements

Discussion

Methodological

changes

During

the

development

of

only

very

minor

changes

were

made

to

the

set

of

approximations

The

main

change

was

in

the

construction

of

the

training

set

used

for

parameter

optimization

One

of

the

most

important

changes

was

the

use

of

rules

in

the

training

set

to

define

chemical

information

that

was

not

a

function

of

any

single

molecule

In

earlier

methods

the

training

set

had

included

only

standard

reference

data

Of

their

nature

such

data

could

not

allow

for

chemical

facts

that

were

independent

of

any

one

moiety

For

example

the

strength

of

a

hydrogen

bond

is

of

great

importance

in

biochemistry

but

it

could

not

be

expressed

in

terms

of

a

single

species

By

use

of

rules

the

value

of

some

chemical

quantity

could

be

related

to

that

of

another

In

the

case

of

hydrogen

bonding

the

heat

of

formation

of

the

water

dimer

was

made

a

function

of

the

heat

of

formation

of

two

separated

water

molecules

Rules

were

particularly

useful

when

elements

of

the

three

transition

metal

series

were

being

optimized

Many

complexes

of

these

elements

are

highly

labile

and

in

the

early

stages

of

parameter

optimization

there

was

a

strong

tendency

for

the

optimized

geometry

of

such

complexes

to

be

qualitatively

incorrect

Faults

of

this

kind

could

not

be

corrected

by

simply

increasing

the

weight

assigned

to

the

correct

geometry

so

rules

were

developed

to

indicate

that

the

faulty

geometries

were

indeed

incorrect

Specific

points

on

the

potential

energy

surface

were

selected

and

from

single-point

high

level

calculations

the

relative

energy

of

these

points

above

the

minimum

was

evaluated

The

points

selected

were

precisely

those

qualitatively

incorrect

geometries

resulting

from

the

use

of

the

then-current

parameters

The

fact

that

the

incorrect

geometry

was

predicted

by

high

level

methods

to

be

of

higher

energy

than

the

correct

geometry

was

then

added

to

the

set

of

rules

A

good

example

of

such

a

rule

was

the

rule

concerning

Fe

-LRB-

CO

-RRB-

mentioned

above

in

which

the

only

datum

that

was

defined

referred

to

the

relative

energies

of

the

compound

in

two

different

symmetries

No

reference

was

made

to

the

bond

lengths

or

bond

angles

With

such

a

rule

in

place

the

parameters

could

be

re-optimized

to

minimize

the

error

arising

from

the

rule

with

the

effect

that

the

energy

of

the

incorrect

symmetry

increased

relative

to

that

of

the

correct

symmetry

In

the

majority

of

cases

one

rule

of

this

type

was

sufficient

less

frequently

two

rules

were

used

and

in

rare

cases

even

more

rules

were

necessary

Another

change

was

the

use

of

very

large

reference

data

training

sets

In

earlier

parameterizations

the

training

set

used

was

deliberately

made

as

small

as

possible

Only

when

the

resulting

method

was

used

in

a

survey

of

species

not

used

in

the

training

set

could

the

predictive

power

of

the

method

be

determined

The

training

set

used

in

the

development

of

was

designed

to

be

considerably

larger

than

the

survey

set

The

rationale

for

this

was

that

by

including

in

the

training

set

reference

data

for

unconventional

species

e.g.

non-equilibrium

and

hypothetical

species

a

greater

region

of

the

error-function

surface

could

be

defined

This

would

in

turn

result

in

a

better

definition

of

the

values

of

the

parameters

That

this

is

useful

can

be

evidenced

by

the

recent

work

in

parameterizing

chlorine

at

the

level

where

the

compound

trifluoroethane

has

a

reported

ΔHf

of

kcal

mol

but

the

value

predicted

using

was

kcal

mol

That

is

the

value

was

in

error

by

over

kcal

mol

If

this

compound

had

been

included

in

the

training

set

it

is

highly

likely

that

the

error

would

have

been

significantly

reduced

Although

over

reference

data

were

used

in

the

training

set

there

are

several

indications

that

even

this

large

number

is

still

inadequate

for

the

definition

of

the

values

of

the

parameters

and

that

an

even

larger

training

set

would

be

highly

desirable

In

light

of

this

work

has

begun

on

identifying

species

to

be

added

to

the

training

set

During

the

testing

of

several

faults

were

found

in

the

method

Some

of

these

were

quickly

traced

to

specific

core-core

parameters

One

of

the

hydrogen

atoms

in

the

complex

-LSB-

ScIII

-LRB-

-RRB-

-RSB-

was

predicted

to

readily

move

toward

the

central

atom

with

the

result

that

a

Sc-H

bond

was

formed

Such

faults

could

easily

be

corrected

by

the

addition

to

the

training

set

of

appropriate

reference

data

from

high-level

calculations

This

was

done

in

several

instances

and

the

specific

error

was

corrected

but

this

action

then

also

required

all

the

testing

to

be

re-started

Because

this

was

a

time-consuming

process

when

faults

were

found

near

the

end

of

the

testing

phase

the

decision

was

taken

that

the

fault

should

be

noted

as

in

the

Sc-H

error

mentioned

here

and

to

take

no

further

action

at

that

time

A

different

type

of

error

found

only

near

the

end

of

testing

was

the

unrealistically

large

p

electron

population

of

some

transition

metals

The

values

of

the

parameters

that

determine

the

p

population

are

defined

using

two

very

different

groups

of

reference

data

atomic

energy

levels

and

conventional

properties

of

polyatomics

If

atomic

energy

levels

were

excluded

from

the

parameter

optimization

then

the

p

population

would

become

very

small

but

if

atomic

energy

levels

were

excluded

then

the

resulting

method

would

not

be

suitable

for

reproducing

such

levels

The

decision

to

use

all

available

atomic

energy

levels

in

the

training

set

was

a

value

judgement

In

the

next

training

set

it

is

likely

that

the

result

of

this

decision-making

process

will

be

different

Detecting

faults

in

semiempirical

methods

is

difficult

and

rather

than

wait

until

all

errors

of

this

type

were

found

and

fixed

a

process

that

could

potentially

take

several

more

years

the

decision

was

made

to

freeze

the

parameters

at

their

current

value

Obviously

still

has

many

errors

some

have

already

been

described

Work

has

already

started

in

an

attempt

to

correct

them

Elimination

of

computational

artifacts

Earlier

NDDO

methods

particularly

and

produced

artifacts

in

potential

energy

surfaces

as

a

result

of

unrealistic

terms

in

the

core-core

approximation

specifically

in

the

set

of

Gaussian

functions

used

In

only

one

Gaussian-type

correction

to

the

core-core

potential

is

allowed

and

consequently

the

potential

for

these

artifacts

has

been

reduced

On

the

other

hand

because

uses

diatomic

parameters

the

likelihood

of

readily-characterized

errors

involving

specific

pairs

of

atoms

e.g.

Sc

and

H

as

mentioned

earlier

is

increased

Errors

of

this

type

can

be

easily

eliminated

by

a

re-parameterization

of

the

faulty

diatomic

There

are

over

sets

of

diatomic

interactions

parameterized

in

covering

most

of

the

common

types

of

chemical

bonds

But

the

number

of

potential

bonds

is

much

larger

given

elements

there

are

almost

diatomic

sets

If

a

molecule

contains

two

elements

for

which

the

diatomic

interaction

parameters

are

missing

then

provided

the

elements

are

well

separated

say

by

more

than

Ångstroms

the

absence

of

the

parameters

will

not

be

important

If

the

two

elements

were

near

to

each

other

then

the

diatomic

core-core

parameters

would

be

needed

This

would

involve

generating

a

small

training

set

of

reference

data

that

included

a

few

examples

of

the

type

of

interaction

involved

and

optimizing

the

two

terms

in

the

diatomic

interaction

This

ability

to

add

diatomic

parameter

sets

to

without

modifying

the

underlying

parameterization

has

the

advantage

that

more

and

more

types

of

interaction

can

be

added

without

changing

the

essential

nature

of

the

method

Accuracy

being

the

most

recent

member

of

the

NDDO

family

of

approximate

semiempirical

methods

is

understandably

the

most

accurate

The

development

of

each

new

method

has

been

guided

by

the

knowledge

of

the

documented

faults

found

in

the

earlier

methods

This

is

reflected

in

the

steady

decrease

in

AUE

of

simple

organic

compounds

from

kcal

mol

for

to

kcal

mol

for

Several

low-energy

phenomena

are

predicted

more

accurately

by

with

the

most

important

of

these

being

the

prediction

of

the

energies

and

geometries

involved

in

hydrogen

bonding

One

consequence

of

this

increased

accuracy

is

that

the

lowest

energy

conformer

of

acetylacetone

is

now

correctly

predicted

to

be

the

ene-ol

structure

and

not

the

twisted

di-one

configuration

Despite

the

improvement

in

hydrogen

bonding

a

significant

error

was

found

in

the

balance

of

energies

involved

in

forming

zwitterions

of

hydroxyl

and

amine

groups

This

is

best

illustrated

by

the

dimer

of

where

predicts

that

the

zwitterion

should

be

kcal

mol

more

stable

than

the

neutral

form

but

higher

level

calculations

indicate

that

the

neutral

form

should

be

kcal

mol

more

stable

than

the

zwitterion

In

the

solid

state

CSD

entries

-LSB-

-RSB-

exists

as

the

neutral

species

In

general

however

average

unsigned

errors

in

ΔHf

have

steadily

decreased

as

semiempirical

methods

have

evolved

Earlier

NDDO

methods

such

as

and

had

AUE

significantly

larger

than

the

Hartree

Fock

method

With

the

advent

of

and

errors

were

intermediate

between

HF

and

In

the

current

work

AUE

in

ΔHf

are

lower

than

those

of

both

and

HF

This

increase

in

accuracy

of

prediction

of

ΔHf

relative

to

higher

level

methods

should

not

be

construed

as

disparaging

those

methods

semiempirical

methods

in

general

and

in

particular

were

parameterized

to

reproduce

ΔHf

The

performance

of

these

methods

when

applied

to

non-equilibrium

systems

in

particular

transition

states

is

likely

to

be

very

inferior

to

that

of

or

HF

As

a

result

of

the

current

work

there

is

a

clear

strategy

for

further

improving

the

accuracy

of

semiempirical

methods

All

three

potential

sources

of

error

need

to

be

addressed

Regarding

reference

data

considerably

more

data

are

needed

than

were

used

here

This

would

likely

come

from

increased

use

of

high-level

theoretical

methods

methods

significantly

more

accurate

than

those

used

here

would

obviously

be

needed

in

any

future

work

Parameter

optimization

can

be

performed

with

confidence

and

reliability

particularly

when

well-behaved

systems

are

used

In

all

cases

examined

where

problems

were

encountered

in

parameter

optimization

problems

also

occurred

in

the

normal

SCF

calculation

in

This

implies

that

as

faults

in

the

SCF

procedure

are

corrected

faults

in

parameter

optimization

would

also

be

removed

Permanent

errors

Notwithstanding

the

optimism

just

expressed

not

all

errors

can

be

eliminated

by

better

data

and

better

optimizations

Despite

strenuous

efforts

some

calculated

quantities

persistently

failed

to

agree

with

the

reference

values

Many

potential

causes

for

these

failures

were

investigated

In

each

case

the

weight

for

the

offending

quantity

was

increased

considerably

and

the

parameter

optimization

re-run

When

that

was

done

the

specific

error

decreased

but

errors

elsewhere

increased

disproportionately

Since

the

final

gradient

of

the

error

function

was

acceptably

small

it

followed

that

the

parameter

optimization

was

not

in

error

The

reference

data

were

checked

to

ensure

that

they

were

in

fact

trustworthy

Because

two

of

the

three

possible

origins

of

error

had

been

eliminated

the

inescapable

conclusion

was

that

there

is

a

fault

in

the

set

of

approximations

The

most

serious

of

these

faults

was

the

qualitatively

incorrect

prediction

of

the

geometry

of

the

exceedingly

simple

system

iron

pentacarbonyl

Conclusions

The

NDDO

method

has

been

modified

by

the

adoption

of

Voityuk

's

core-core

diatomic

interaction

parameters

This

has

resulted

in

a

significant

reduction

in

error

for

compounds

of

main-group

elements

and

together

with

Thiel

's

d-orbital

approximation

allows

extension

of

the

NDDO

method

to

the

whole

of

the

transition

metal

block

The

accuracy

of

in

predicting

heats

of

formation

for

compounds

of

interest

in

biochemistry

is

somewhat

better

than

Hartree

Fock

or

DFT

methods

using

the

-LRB-

d

-RRB-

basis

set

For

a

representative

set

of

compounds

gave

an

average

unsigned

error

of

kcal

mol

for

the

same

set

HF

and

had

AUE

of

and

kcal

mol

respectively

The

potential

exists

for

further

large

increases

in

accuracy

This

would

likely

result

from

the

increased

use

of

accurate

reference

data

derived

from

high-level

methods

and

from

the

development

of

better

tools

for

detecting

errors

at

an

early

stage

of

method

development

Electronic

supplementary

material

Tables

of

errors

in

predicted

heats

of

formation

geometries

dipole

moments

and

ionization

potentials

obtained

using

HF

and

for

individual

species

are

provided

together

with

references

for

all

reference

data

used

These

data

were

used

in

generating

the

statistics

presented

in

the

discussion

on

accuracy

ESM

DOC

MB

nddo
parameterization
pm6
transition metals