Chapter 1 Ab Initio Calculation of Potential Energy Surfaces By R. F. W. Bader and R. A. Gangi, 1, 
Chapter 2 Intermolecular Forces By J. G. Stamper, 66, 
Chapter 3 Quantum Mechanical Calculations on Small Molecules By C. Thomson, 83, 
Chapter 4 Electronic Calculations on Large Molecules By B. J. Duke, 159, 
Author Index,

CHAPTER 1

Ab initio Calculation of Potential Energy Surfaces

BY R. F. W. BADER AND R. A. GANGI

1 The Concept of a Potential Energy Surface

This chapter is concerned with the calculation of potential energy surfaces by non-empirical methods, i.e. by obtaining solutions to Schrödinger's equation within the Born-Oppenheimer approximation. The concept of a potential energy surface is a consequence of the separation of the nuclear and electronic motions as proposed by Born and Oppenheimer. The nuclei may be considered to move under the influence of a potential determined by their mutual electrostatic repulsion and by the total energy of the electrons, an energy which is determined for every possible static configuration of the nuclei. The gain in conceptual simplicity afforded by the Born-Oppenheimer procedure is enormous and its use underlies many of our chemical concepts. For example, energies of activation or energy barriers in general, potential constants, the frequencies of vibrational and rotational motions, and bond lengths and bond angles as determined by an energy minimum, are all concepts defined in terms of the properties of a potential surface.

To obtain information concerning a system of n electrons and N nuclei in the absence of external fields, one must solve the time-independent Schrödinger equation

[FORMULA NOT REPRODUCIBLE IN ASCII] (1)

where the properties of the ith stationary state of the system of energy Ei are obtainable from the eigenfunctions Ψi(x,R) which in turn depend upon the space and spin co-ordinates of all the electrons and nuclei (whose collective co-ordinates are denoted by x and R, respectively) in the system. The total Hamiltonian operator is

[FORMULA NOT REPRODUCIBLE IN ASCII] (2)

where [??]e and [??N are the kinetic energy operators of the electrons and nuclei, respectively, and where the potential energy operator in atomic units is,

[FORMULA NOT REPRODUCIBLE IN ASCII] (3)

One can distinguish three levels of approximation in obtaining solutions to equation (1):2 (i) the Born-Oppenheimer approximation, often referred to as the clamped-nucleus approximation; (ii) the adiabatic approximation; and (iii) the non-adiabatic approximation. The solutions to equation (1) provided by the first of these are entirely adequate for most problems of chemical interest. For example, use of this approximation in the calculation of the low-lying vibrational and rotational levels of H+2 yields results which are correct within experimental accuracy.

In the Born-Oppenheimer approximation, one first transforms to a centre-of-mass, molecule-fixed co-ordinate system, thereby yielding a kinetic energy operator for the electrons referred to nuclei that are fixed with respect to their centre of mass and a nuclear kinetic energy operator which refers only to their internal motions, i.e. their vibrations and rotations and the various couplings between them.* Because of the large disparity in the masses of the electrons and the nuclei, the average kinetic energy of the former will be many times that of the latter, the ratio being proportional to mk/me ~ 2000 in the least favourable case when mk refers to the mass of a proton. Thus, in the limiting classical case, one obtains a picture of the electrons undergoing very rapid motion relative to the nuclei and adjusting almost instantaneously to changes in the nuclear positions. This suggests that the electronic motions are determined to a good approximation by just the static field of the nuclei, i.e. the electronic motion is determined by where the nuclei are but not by how fast they are moving. The motions of the nuclei in this separation of co-ordinates are governed by a potential whose negative gradient is simply the electrostatic force of repulsion between the nuclei and the attractive force exerted on the nuclei by the electronic charge distribution, a distribution whose form is determined by the electronic wavefunction evaluated for each configuration of the nuclei.

Mathematically, this separation of the kinetic motions of the electrons and nuclei amounts to approximating the total wavefunction Ψi (x, R) by a simple product of electronic and nuclear wavefunctions

[FORMULA NOT REPRODUCIBLE IN ASCII] (4)

The electronic function y>i(x; R) is obtained by solving the electronic Schrödinger equation for a fixed nuclear configuration,

[FORMULA NOT REPRODUCIBLE IN ASCII] (5)

where

[FORMULA NOT REPRODUCIBLE IN ASCII] (6)

The use of the semi-colon in the notation Ψi(x; R) is to denote the explicit dependence of Ψi on the electronic space-spin co-ordinates x and its implicit dependence, along with Ei on the nuclear co-ordinates R. Clearly, the solution of equation (5) for all spatial arrangements of the nuclei will generate an energy (hyper)-surface, the potential energy surface, which governs the motion of the nuclei as obtained by solving the nuclear eigenvalue equation

[FORMULA NOT REPRODUCIBLE IN ASCII] (7)

Equation (7) and its implied assumption of the separability of the electronic and nuclear motions is called the Born-Oppenheimer approximation.

The exact Schrödinger equation of motion, equation (1), may be equivalently stated in a manner which shows the neglected terms arising from the assumption of the product form for the wavefunction, equation (4). The exact eigenstate Ψi(x, R) is expanded in terms of the complete orthonormal set of functions Ψi(x; R) obtained from the solutions of the electronic equation, equation (5), in which case the nuclear wavefunctions χi (R) appear as the coefficients in the expansion. This procedure yields the following infinite set of coupled equations for the χi(R)

[FORMULA NOT REPRODUCIBLE IN ASCII] (8)

From this formal, but exact statement of Schrödinger's equation, which corresponds to the non-adiabatic approximation referred to above, one sees that the approximation made in obtaining equation (7) is to assume that

[FORMULA NOT REPRODUCIBLE IN ASCII] (9)

for all i and j. In another order of approximation, with the expansion of Ψi(x, R) truncated to the single product term as given in equation (4), one obtains in place of equation (8),

[FORMULA NOT REPRODUCIBLE IN ASCII] (10)

for the nuclear eigenfunction χiv(R). The term <Ψi|[??]NΨi> is called the adiabatic correction, a term which, as argued above, should contribute only 1/2000 or less than does Ei(R) to the total energy of the system.

Ko[??]os and Wolniewicz have obtained extremely accurate solutions to Schrödinger's electronic equation, equation (5), for H2 in its ground state over a large range of intemuclear distances. They further obtained solutions...

„Über diesen Titel“ gehört möglicherweise zu einer anderen Auflage dieses Titels.

PBShop.store US

Wood Dale, IL, USA

Verkäufer/-in mit 5 Sternen

Verkäufer:in bei ZVAB seit 7. April 2005

Versandkosten innerhalb von USA

Artikel	7 bis 14 Werktage	7 bis 14 Werktage
Erster Artikel	EUR 0,00	EUR 0,00

Die Versandzeiten werden von den Verkäuferinnen und Verkäufern festgelegt. Sie variieren je nach Versanddienstleister und Standort. Sendungen, die den Zoll passieren, können Verzögerungen unterliegen. Eventuell anfallende Abgaben oder Gebühren sind von der Käuferin bzw. dem Käufer zu tragen. Die Verkäuferin bzw. der Verkäufer kann Sie bezüglich zusätzlicher Versandkosten kontaktieren, um einen möglichen Anstieg der Versandkosten für Ihre Artikel auszugleichen.

Zahlungsarten

Shopbeschreibung

We first started out as “The Paperback Exchange,” a chain of physical bookstores where we would part exchange your beloved books for new stories to transport you to faraway places. However, as shopping started to evolve to online shops and marketplaces, we bid our stores goodbye to become “PBShop.” This transition has only allowed us to blossom as we now ship thousands of titles to book lovers across the globe. We pride ourselves in being a community of local book lovers which allows our passion and devotion to shine in everything we do. In 2020 we not only celebrated our 20th birthday but our 1st birthday as being completely employee owned after becoming an E.O.T in September 2019. We are proud to be different and embrace standing apart on a book mountain by working from a virtual inventory which allows us to provide thousands of books that may be difficult to get for your bookshelf or your studies. Working with a number of different suppliers allows us to explore other avenues such as puzzles, sheet music and even stationery so we really do have something for everyone. Life is about being versatile in all realms of existence. If this is your first visit or you are a returning customer, we would like to welcome you to the PBShop family, for there is no friend as loyal as a book. We are a company who put our customers at the centre of everything we do as we understand the importance of reading because once you learn to read, you will forever be free. There are a whole lot of things in this world of ours that we are yet to explore, which is why we will forever inspire curious minds.…

Spezialisierung

Hardbacks, Paperbacks

Unternehmensdaten der Verkäuferin bzw. des Verkäufers

Pbshop.co.uk Ltd

Unit 22 Horcott Industrial Estate, Horcott Road
Fairford, Vereinigtes Königreich GL7 4BX

Verkäufer/-in kontaktierenArtikel dieser Verkäuferin bzw. dieses Verkäufers anzeigenHomepage des Verkäufers anzeigen

Verkaufsbedingungen

Returns Policy
We ask all customers to contact us for authorisation should they wish to return their order. Orders returned without authorisation may not be credited.
If you wish to return, please contact us within 14 days of receiving your order to obtain authorisation.

Returns requested beyond this time will not be authorised.

Our team will provide full instructions on how to return your order and once received our returns department will process your refund.
Please note the cost to return any unwanted order to us is borne by the buyer.

Should your order arrive damaged, not as advertised or faulty, we must be advised of this within 14 days of delivery. Please contact us so we can find the best solution for you.

Our Customer Care Team can be contacted via emailing paperback-us@paperbackshop.co.uk, or by calling our UK Office on +441285 712 917. We are available 9:00am till 5:30pm GMT weekdays and 9:00am till 1:00pm GMT on Saturdays.Returns Policy
We ask all customers to contact us for authorisation should they wish to return their order. Orders returned without authorisation may not be credited.
If you wish to return, please contact us within 14 days of receiving your order to obtain authorisation.
Returns requested beyond this time will not be authorised.
Our team will provide full instructions on how to return your order and once received our returns department will process your refund.
Please note the cost to return any unwanted order to us is borne by the buyer.
Should your order arrive damaged, not as advertised or faulty, we must be advised of this within 14 days of delivery. Please contact us so we can find the best solution for you.
Our Customer Care Team can be contacted via emailing paperback-us@paperbackshop.co.uk, or by calling our UK Office on +441285 712 917. We are available 9:00am till 5:30pm GMT weekdays and 9:00am till 1:00pm GMT on Saturdays.

Widerrufsrecht

Wenn Sie Verbraucher sind, können Sie gemäß den folgenden Bestimmungen vom Vertrag zurücktreten. Verbraucher ist jede natürliche Person, die zu Zwecken handelt, die nicht ihrer kaufmännischen, gewerblichen, künstlerischen oder beruflichen Tätigkeit zugerechnet werden können.

Informationen zum Widerrufsrecht

Gesetzliches Widerrufsrecht

Sie haben das Recht, den Vertrag innerhalb von 14 Tagen ohne Angabe von Gründen zu widerrufen.

Die Widerrufsfrist beträgt 14 Tage ab dem Tag, an dem Sie oder ein von Ihnen benannter Dritter, der nicht der Transporteur ist, die letzte Ware oder den letzten Posten oder das letzte Exemplar in Besitz genommen hat.

Um das Widerrufsrecht auszuüben, füllen Sie auf unserer Website unter „Meine Einkäufe" in „Mein Nutzerkonto" eine eindeutige Erklärung elektronisch aus und senden Sie sie ab. Wir werden Ihnen unverzüglich eine Bestätigung über den Eingang eines solchen Widerrufs auf einem dauerhaften Datenträger (z. B. per E-Mail) übermitteln.

Um die Widerrufsfrist einzuhalten, reicht es aus, dass Sie Ihre Mitteilung über die Ausübung des Widerrufsrechts vor Ablauf der Widerrufsfrist absenden.

Auswirkungen des Widerrufs

Wenn Sie diesen Vertrag widerrufen, erstatten wir Ihnen alle Zahlungen, die wir von Ihnen erhalten haben, einschließlich der Lieferkosten (mit Ausnahme der zusätzlichen Kosten, die entstehen, wenn Sie eine andere Art der Lieferung als die von uns angebotene günstigste Standardlieferung gewählt haben).

Wir können einen Abzug von der Rückerstattung für den Wertverlust der gelieferten Waren vornehmen, wenn der Verlust auf eine unnötige Behandlung durch Sie zurückzuführen ist.

Wir werden die Rückerstattung unverzüglich und nicht später als 14 Tage nach dem Tag vornehmen, an dem wir über Ihre Entscheidung, diesen Vertrag zu widerrufen, informiert wurden.

Für die Rückerstattung verwenden wir dasselbe Zahlungsmittel, das Sie für die ursprüngliche Transaktion verwendet haben, es sei denn, Sie haben ausdrücklich etwas anderes vereinbart; in keinem Fall werden Ihnen aufgrund einer solchen Rückerstattung Gebühren berechnet.

Wir können die Rückzahlung verweigern, bis wir die Waren wieder zurückerhalten haben oder Sie den Nachweis erbracht haben, dass Sie die Waren zurückgesandt haben, je nachdem, was eher eintritt.

Sie müssen die Waren unverzüglich und in jedem Fall spätestens 14 Tage ab dem Tag, an dem Sie uns über den Widerruf dieses Vertrags unterrichten, an PBShop.store US, Springfield Gardens, New York, U.S.A., +44 1285 712917, zurücksenden oder übergeben. Die Frist ist eingehalten, wenn Sie die Ware vor Ablauf der Frist von 14 Tagen zurücksenden. Sie müssen die direkten Kosten der Rücksendung der Waren tragen. Sie haften nur für einen etwaigen Wertverlust der Waren, der auf eine Behandlung zurückzuführen ist, die nicht zur Prüfung der Art, Eigenschaften und Funktionsweise der Waren erforderlich ist.

Ausnahmen vom Widerrufsrecht

Das Widerrufsrecht gilt nicht für:

Die Lieferung von Zeitungen, Zeitschriften oder Magazinen mit Ausnahme von Abonnementverträgen; und
Die Lieferung digitaler Inhalte, die nicht auf einem physischen Medium (z. B. auf einer CD oder DVD) geliefert werden, wenn Sie bei Ihrer Bestellung akzeptiert haben, dass wir mit der Lieferung beginnen können und dass Sie nach Beginn der Lieferung den Vertrag nicht mehr widerrufen können.

Versandbedingungen

Books are shipped from our US or UK warehouses. Delivery estimates allow for delivery from either location.

Datenschutz und Sicherheit

We are committed to protecting the privacy and security of personal data. This policy tells you how we will collect and use your personal data, and what you should expect in respect of the personal information about you that we have collected.

We are PBShop.co.uk Limited (registered in England & Wales with Company No: 9366362). Our registered office is at Swatton Barn, Badbury, Swindon, SN4 0EU, United Kingdom.

The person with responsibility for our data protection compliance is the Managing Director and they can be contacted via email at help@pbshop.store.

We are the data controller of the personal data that we keep and use, and we are therefore responsible for making sure that our systems, processes and people comply with the relevant data protection laws in respect of that personal data.

We will act in respect of personal data to comply with the six principles of the General Data Protection Regulation (“GDPR”), which are:

Lawfulness, fairness and transparency;
Purpose limitation;
Data minimisation;
Accuracy;
Storage limitation;
Integrity and confidentiality.
You have rights in respect of how your personal data can be processed; these include the right to request:

a copy of your personal data;
that inaccurate data is rectified; and
that your personal data is, in certain circumstances, erased or restricted.
You have the right to complain to the Information Commissioner, which you can do by contacting the Information Commissioner’s Office (ICO) directly. Full contact details, including a helpline number, can be found on the ICO website (www.ico.org.uk). This website also has useful information on your rights and our obligations. However, please raise any concerns or issues with us first so that we may deal with this as quickly as possible for you.

COLLECTION, USE AND DISCLOSURE OF PERSONAL DATA
We collect and process data for the following reasons:

personal data collected and created in relation to our selling books and other items; and
personal data relating to our people (which means those people working for our company, or providing services to us, or potentially working or providing services to us, including employees, consultants, temporary or casual workers and contractors).
All of our people are required to abide by our Privacy Policy when handling personal data, and are provided with appropriate data protection training. Any breach of data protection will be taken seriously, and may result in disciplinary action.

Selling books and other items
COLLECTION:
We will collect personal information which may include names, addresses and other information for our customers. This will be done for the management of the commercial relationship with our customers.

We may collect this personal data from you directly, and also from third parties, including publicly available third party sources. Additional personal data will be collected as it is created by or supplied to us in the course of anything we do for you.

USE:
We will use the personal data in the course of making sales and to comply with requirements that we are required to undertake.

We will only use your personal data when the law allows us to. Most commonly, we will use your personal data in the following circumstances:

in performance of a contract (the agreement to sell books and other items); and/or
compliance with a legal obligation; and/or
vital interests of the data subject; and/or
where it is necessary for our legitimate interests (or those of a third party) and your interests and fundamental rights do not override those interests.
We may also use your personal data in the following situations, which are likely to be far less common:

where we need to protect your interests (or someone else’s interests);
where it is needed in the public interest or for official purposes.
DISCLOSURE:
We may share personal data with third parties as is necessary in the sale of books and other items, including to third party providers who deliver services to support the operation of our company.

In respect of all disclosures of personal data, we will only share the personal information which is necessary for the particular purpose for which it is provided, or where we have another legitimate interest in doing so, and we will ensure that the personal data is appropriately protected.

RETENTION PERIOD:
We will keep personal data only for as long as is necessary to fulfil the purposes for which we collected it, including for the purposes of satisfying any legal, accounting or reporting requirements. Normally, our retention period for personal data collected for this purpose is a minimum of 6 years after the end of the period that we are selling books and other items.

Our people
COLLECTION:
We will collect names, addresses, contact details, education and employment history, identity and other background checks (which may include appropriate criminal and financial checks and confirmation of the right to work in the UK), marital status and information on next of kin and dependents, financial information (such as bank details and NI numbers), date of birth, gender, a copy of your passport and driving licence, performance information and compensation history, recruitment information (including references), salary, benefits, pension and annual leave information, disciplinary and grievance information, photographs, information about your use of our information and communications systems and CCTV footage or other information obtained through electronic means.

We may collect and use special categories of personal data as required in the carrying out of obligations and exercising specific rights of us or the data subject in the field of employment. This may include information relating to matters such as health, racial origin, religious belief and offences or alleged offences.

Personal data may be collected from you during the selection process (for example via your application form and CV) and during your employment (including holiday forms, expense claims, performance reviews, any disciplinary or grievance processes), or from third parties, including referees, health service providers, background check providers.

USE:
Personal data of our people will be used for HR administration and management, both in respect of the selection of people to work for us (including suitability, eligibility and/or fitness to work), and those who do work for us, to include learning and development, disciplinary and security (of people, offices and data) requirements, providing and liaising with benefits providers, business management and planning (including accounting and auditing), paying you and dealing with tax and NI deductions, assessing and deciding on salary reviews and compensation, conducting performance reviews and managing performance, dealing with legal disputes involving you or others, preventing fraud, monitoring your use of our information and communications systems, ensuring compliance with our policies, and equal opportunities monitoring.

We will only use your personal data when the law allows us to. Most commonly, we will use your personal data in the following circumstances:

In performance of a contract with you; and/or
compliance with a legal obligation; and/or
vital interests of the data subject; and/or
where it is necessary for our legitimate interests (or those of a third party) and your interests and fundamental rights do not override those interests.
We may also use your personal data in the following situations, which are likely to be less common:

where we need to protect your interests (or someone else’s interests);
where it is needed in the public interest or for official purposes.
In respect of all disclosures of personal data, we will only share the personal information which is necessary for the particular purpose for which it is provided, or where we have another legitimate interest in doing so, and we will ensure that the personal data is appropriately protected.

In respect of special categories of personal data, we may process this in the following circumstances:

in limited circumstances with your explicit written consent; and/or
where we need to carry out our legal obligations and in line with our policies; and/or
where it is needed in the public interest (such as for equal opportunities monitoring or in relation to our occupational pension scheme), and in line with our policies; and/or
where it is needed to assess your working capacity on health grounds, subject to appropriate confidentiality safeguards.
Less commonly, we may process this type of information where it is needed in relation to legal claims or where it is needed to protect your (or someone else’s) interests and you are not capable of giving your consent, or where you have already made the information public. We may also process such information about our people (or former people) in the course of our legitimate business activities with the appropriate safeguards.

The special categories of personal data may be used in the following ways:

in relation to leaves of absence, to comply with employment and other laws;
in relation to your physical or mental health, or disability status, to ensure your health and safety in the workplace and to assess your fitness to work, to monitor, manage and administer benefits and absences;
in relation to your race, national or ethnic origin, religious, philosophical or moral beliefs, or your sexual life or sexual orientation, to ensure meaningful equal opportunity monitoring and reporting.
DISCLOSURE:
Personal data may be transferred to service providers who support the operation of our business (such as payroll service providers), to other third parties reasonably necessary in the conduct of our business (including insurers, professional advisors, regulators). These third parties may be acting as processor, or controller of personal data in their own right.

RETENTION PERIOD:
We will keep personal data only for as long as is necessary to fulfil the purposes for which we collected it, including for the purposes of satisfying any legal, accounting or reporting requirements. We will normally keep personal data of applicants who we do not employ for six months after we receive it.

We will keep personal data of employees throughout your employment and normally for a minimum of six years after your employment ends.

Changes to your personal data
It is very important that the personal information that we hold about you is accurate and current. Please tell us if your personal information changes during your relationship with us.

Data Security
We have put in place measures to prevent your personal data from being accidentally lost, used or accessed in an unauthorised way, altered or disclosed. In addition, we limit access to your personal data to those of our people and other third parties who have a business need to know. They will only process your personal data on our instructions and where they have agreed to treat the information confidentially and to keep it secure. We have put in place procedures to deal with any suspected data security breach and will notify you and the ICO of a suspected breach where we are legally required to do so.

Where we store your personal data
We principally store data, both electronically and on file, at our office.

Personal data may be transferred outside of the EEA by processors acting on our behalf. For transfers to countries not considered adequate by the EC, we will ensure that personal data is adequately protected, as required by the GDPR. This would include by use of the Standard Contractual Clauses adopted by the EC to protect personal data.

Your rights
Under certain circumstances, you have the right by law to request:

access to your personal data. This enables you to ask to receive a copy of the personal data that we hold about you and to check that we are lawfully processing it.
correction of the personal data that we hold about you.
erasure of your personal data.
object to processing of your personal data where we are relying on our legitimate interest and there is something about your particular situation which makes you want to object to processing on this ground. You also have the right to object where we are processing your personal information for direct marketing purposes.
restriction of processing of your personal data.
transfer of your personal data to another party.
Links from our website
Our website may, from time to time, contain links to and from the websites of third parties that we permit to make such links. If you follow a link to any of these websites, please note that these websites have their own privacy policies and that we do not accept any responsibility or liability for these policies. We recommend that you check these policies before you submit any personal data to these websites.

CHANGES TO THIS PRIVACY POLICY
Any changes we may make to our privacy policy in the future will be posted on our website, so please ensure that you are viewing the correct version.

Please contact us if you have any questions, comments or requests regarding this Privacy Policy.

Last amended May 2021.

Theoretical Chemistry

Artikelbeschreibung des Verkäufers

Bibliografische Details

Inhaltsangabe

Über diesen Titel

Auszug. © Genehmigter Nachdruck. Alle Rechte vorbehalten.

Theoretical Chemistry Volume 2

A Review of the Recent Literature

The Royal Society of Chemistry

Contents