6: Sets

  • 6.1: Set Notation
    A set is just a collection of items and there are different ways of representing a set. We want to be able to both read the various ways and be able to write down the representation ourselves in order to best display the set. We have already seen how to represent a set on a number line, but that can be cumbersome, especially if we want to just use a keyboard.
  • 6.2: The Complement of a Set
    Complements come up very often in statistics, so it is worth revisiting this, but instead of graphically we will focus on set notation. Recall that the complement of a set is everything that is not in that set. Sometimes it is much easier to find the probability of a complement than of the original set, and there is an easy relationship between the probability of an event happening and the probability of the complement of that event happening.
  • 6.3: The Union and Intersection of Two Sets
    All statistics classes include questions about probabilities involving the union and intersections of sets. In English, we use the words "Or", and "And" to describe these concepts. In this section we will learn how to decipher these types of sentences and will learn about the meaning of unions and intersections.
  • 6.4: Venn Diagrams
    Venn Diagrams are a simple way of visualizing how sets interact. Many times we will see a long wordy sentence that describes a numerical situation, but it is a challenge to understand. As the saying goes, "A picture is worth a thousand words." In particular a Venn Diagram describes how many elements are in each set displayed and how many elements are in their intersections and complements.

6 Sets to Build Swimming Endurance

Along with swimming smooth, swimming with sustainability, or swimming endurance, is the most important skill for a triathlete to have. It does a triathlete no good to get out in front of the pack, swim strong to the first buoy, then completely come apart and struggle the rest of the way. The ability to sustain a high intensity is paramount, behind only smooth in importance. Having a pretty stroke for half the race and then watching your arms come off and float to the bottom of the ocean isn't terribly useful.

As stated, a major part of sustainable work will be focused on maintaining smooth swimming. The other focus of sustainable swimming is being fit enough to allow you to get out of the water after 1.5K and blast up the beach, through T1 and out onto your bike.

These will be longer sets, much like your long, slow-distance runs and rides. Maintaining a steady pace and heart rate is the goal, not cranking it to 11.

Be sure to warm up before you go into your main sets with 200 to 500 yards nice and easy. This should shake the cobwebs out and get the blood into your muscles. You can also use active rest, 50 to 100 yards easy.

Please note that under set #2 there are many ideas for drill variation. Be creative with this. These are guidelines and ideas. They can be used on almost any of these sets. Many of the other sets also have variation possibilities.

Full-Body Workouts of the Legends

Full body training is vastly underappreciated. Too bad, because it works exceptionally well.

Consider that, occasionally, life happens and there are times when you can only hit the gym two or three times a week. Those on five to six day splits often end up missing workouts because of work or family obligations. That kind of interruption in a split can mean that body parts may go a week or more without stimulation.

Not good! These kinds of scenarios are where full body workouts shine. But even if you’re simply looking for a change of pace, they won’t disappoint.

Weirdly, full body training hasn’t been mainstream for over 50 years, so let’s do a quick refresher.

Who Trains Full Body?

If you asked this question 60 years ago when men like Reg Park and Leroy Colbert were gracing magazines, the better question would be, “Who doesn’t train full body?”

The basic idea was simple – train, recover, and repeat. Guys like Colbert and Park were writing about the philosophy in every major magazine, but let’s look at a guy who didn’t get nearly as much press or fame: George Eiferman.

George Elferman

Eiferman was an actor, stuntman, and Mr. Universe winner who traveled the country teaching the value of physical training to high school students.

He was a huge advocate of full body training and like other champions of the time, it was his go-to methodology. This was one of his favorite full-body, three times a week routines:

Exercise Sets Reps
A Hack Squat 3 7-10
B Bench Press 3 7-10
C Dumbbell Fly 3 7-10
D Dumbbell Lateral Raise 3 7-10
E Alternate Dumbbell Press 3 7-10
F Cheating One-Arm Row 3 7-10
G Cheating Barbell Curl 3 7-10
H Dumbbell Concentration Curl 3 7-10
I Dumbbell Wrist Curl 3 7-10
J Side Bend 3 7-10
K Sit Up 3 8-12

Notice that George did 6 sets apiece for chest, triceps, biceps, and shoulders. That number shows up a lot in the following routines.

Schwarzenegger’s “Golden Six”

While Arnold Schwarzenegger was, at times, the definitive split king, he started his career on a full body routine and he still recommends it for new trainees. The program he used was named “The Golden Six”:

The Golden Six

Exercise Sets Reps
A Barbell Squat 4 10
B Wide-Grip Barbell Bench Press 3 10
C Chin-Up 3 *
D Behind-the-Neck Overhead Press 4 10
E Barbell Curl 3 10
F Bent Knee Sit-Up 3-4 failure

* As many reps as you can for 3 sets

And that’s it. Simple, direct, and effective. Like most full body workouts, he recommended doing them three times a week on alternating days with no weight training performed on rest days. Because of the low volume employed, this type of full body routine is ideal for beginners and those solely looking for strength gains.

Leroy Colbert’s Method: 6 Sets Each for a Total of 42 Sets

Bodybuilding legend Leroy Colbert was very adamant about full-body training and said he never trained productively any other way.

In Colbert’s time, the norm was 3 sets per body part. He explained that after 3 sets, lifters would often drop the weights like they were on fire because they believed that doing more than 3 would make their muscles shrink. Nevertheless, Colbert went against the grain, started doing 6 sets per body part, and the rest is history.

I’ve successfully recommended this type of routine to dozens of lifters and used Colbert’s method exclusively to gain 16 pounds of mostly lean bodyweight while keeping my waist the exact same size. Here are his simple guidelines:

  • Train your entire body every other day.
  • Keep the reps between 6 to 10. If you get 10 reps on all sets, increase the weight.
  • Perform at least 6 sets per body part for optimal growth. You can, however, perform more sets on certain body parts.
  • Establish a mini push/pull setup within the workout. As such, don’t train chest right after triceps or biceps directly after back. For example, training the body in this order fulfills this requirement: Triceps, biceps, chest, back, thighs, shoulders, and finally, calves.
  • Do 6 sets each for a total of 42 sets.

Note: If you think 42 sets is too much, consider what Arnold’s mentor, Reg Park, once said: “I realize I was doing about 90 sets a workout, which I am sure was far more than any British bodybuilder was doing in 48/49. By March/April 1949, I entered and won the Mr. N.E Britain, beating the previous national winner. Not bad for only 7 or 8 months of serious training.”

The basic idea of Colbert’s routine is to have a list of exercises for each muscle group you cycle through (2 per training day), increase the weight whenever possible, and get plenty of rest and proper nutrition.

For example, if you wanted to prioritize your arms, your routine for a particular day might look something like this:

Exercise Sets Reps
A1 Body Drag Curl 4 6-10
A2 Close Grip Bench 4 6-10
B1 Incline Curl 4 6-10
B2 Overhead Triceps Extension 4 6-10
C1 Weighted Pull-up 3 6-10
C2 Incline Bench Press 3 6-10
D1 Bent-Over Barbell Row 3 6-10
D2 Decline Dumbbell Fly 3 6-10
E1 Barbell Front Squat 3 6-10
E2 Military Press 3 6-10
F1 Hack Squat 3 6-10
F2 Dumbbell Lateral Raise 3 6-10
G Standing Calf Raise 3 12-15
H Seated Calf Raise 3 12-15

Total Sets:&emsp46
Rest Between Sets:&emsp1 minute
Set Length:&emsp32 seconds (4 seconds per rep)
Total Time Per Workout:&emsp1 hour and 10 minutes

This routine stimulates every major muscle three times a week. Moreover, it’s a very flexible routine and you should be able to make it your own and reap maximum benefits with the tips below.

Make Colbert’s Routine Your Own

  • Take advantage of the fact you’re training each body part 3 times a week by varying the rep ranges. For example, if strength is your primary concern, then do 5 sets of 5 on two of the days.
  • If you have a favorite exercise that consistently gives you gains, by all means put it as one of your 5 alternates. On the other hand, if something like behind the neck press causes you pain, knock it off the list immediately.
  • Full-body training is very effective at bringing up weak body parts because you can hit them first three times a week.

Gironda’s 8 x 8

Vince Gironda’s 8 x 8 is an outstanding method of training because it allows you to hit every body part three times a week in a short amount of time. It also introduces a different progression method that 99% of trainees have probably never tried. The basics are simple:

  • Perform 8 sets of 8 reps per body part every session.
  • Train 3 times a week.
  • Employ a mini push/pull order within the workout, similar to Leroy Colbert’s method.
  • Train your weakest body parts first.

The progression is where it gets interesting. Instead of simply adding more weight to the bar, you reduce rest times. You start with 45 or 60 seconds of rest between sets and every time you successfully get all 8 reps for all 8 sets, you cut the rest period by 5 seconds.

The starting weight should be around 60% of your normal 8-rep maximum (that’s assuming you normally rest a minute or two for 3-5 sets). The lowest you reduce your rest to is 15 seconds between sets (some may want to stop at 30). At that point, you’d add weight and start over.

You should steadily progress for about 6-8 weeks as this is a different type of overload than your body is used to. By the 8-week mark, some trainees can use 100% of their original 8-rep max for 8 sets of 8 with only 15-30 seconds rest between sets.

However, even if you manage to work your way back to 75% of your original 8-rep max, you’re still doing a lot more work in a lot less time. The pump and the mind muscle connection is incredible and it’s also easy on the joints.

A basic routine might look like this:

Exercise Sets Reps
A Bench Press 8 8
B Lat Pulldown (do pull-ups if you’re a stud) 8 8
C Overhead Triceps Extension 8 8
D Body Drag Curl 8 8
E Lateral Raise 8 8
F Hack Squat 8 8
G Standing Calf Raise 8 20

That’s 56 sets, which would take a little over an hour if using 45-second rest periods, or only 43 minutes if using 15-second rest periods.

The last thing to consider is exercise variety. Obviously you wouldn’t expect to get a thicker and wider back from just doing lat pulldowns for 8 weeks.

You could alternate between two exercises such as pulldowns and bent-over rows and still follow the prescribed progression. The same can be said for any body part you want to practice a variety of movements.

6" Dado Sets

Bigger isn't always better. Do the math: 6" stacked dado sets cost about $20 to $30 less than comparable 8" models, and in most cases they'll do everything their big brothers will. We seldom cut dadoes deeper than 3 ⁄4 ", and all 13 of the 6" sets we tested beat that by 1 ⁄2 ". Also, if you own a lower-powered tablesaw--particularly a benchtop model—a 6" dado set stresses its motor less without sacrificing cut quality or performance.

Before you buy, think about the materials you use most often and the types of cuts you typically make. For example, if you work only with solid wood, ignore how a set performs in plywood and melamine-coated particleboard. But if you frequently dado or rabbet tear-out-prone veneered plywood, select a set that excels at leaving clean edges in that material.

Most 6" dado sets cut cleanly in solid stock, but some leave significant surface tear-out on birch-veneered plywood and melamine-coated particleboard. (In fairness, some manufacturers do not recommend their 6" models for plywood or melamine. Because most of us can afford just one set for doing everything, we tested all sets in all materials to see how they fared against each other.)

If you use a dado set regularly to cut tenons or half-lap joints, choose a model that leaves flat, smooth surfaces, critical for a glue joint. Because each set's outer blades feature beveled carbide teeth (to shear the surface fibers) rather than all flat-teeth (that leave smooth bottoms but increase edge tear-out), these sets all leave tiny scoring grooves at the outer edges of the bottom of each cut. Such grooves don't diminish the joint's strength, but can make visible half-lap joints less attractive.

These sets typically come with at least four chippers of varying thicknesses. (The top-performing sets feature four teeth per chipper instead of two.) You mix and match chippers to achieve the desired width of cut. And often you'll need to adjust the width of your dado stack in tiny increments to get an ideal fit, especially with sheet goods that typically measure less than their advertised thickness. For this adjustment, most 6" dado sets include shims, but some are easier to use than others. For example, some shims come marked with their thicknesses. Other manufacturers color code their shims. One even uses an adjustable hub on one outer blade that allows you to incrementally increase or decrease the width .004" with each click.

Bedroom Furniture Sets

Bedroom Sets by Ashley HomeStore
You know what you like – so why not get an entire bedroom set in one fell swoop to enjoy? Our Ashley Furniture bedroom sets are packed with style, value and variety for trendy bedroom seekers. From opulent tufting to the whitewashed look of shiplap, you’re sure to find the right bedroom set that speaks to your personal tastes. Explore sets with dressers, nightstands and even mattresses – all in the name of giving you the best value, every time.

What Does a Bedroom Set Consist of?
Many of you may be wondering, “Do bedroom sets come with a mattress?” The answer is that some of our sets do! You’ll want to make sure that you read what’s included carefully on our online sets – but a variety of them do have an entire frame and mattress in a box deals. Other bedroom furniture sets, such as a 5-piece, often include an entire bed (with headboard, footboard and rails, totalling 3 parts), a dresser and a mirror.

What Bedroom Set Material is Most Durable?
All of our bedroom sets are built to be durable and stylish. There are many ways you can ensure your bed set lasts: if you move, ensure it’s arranged in a moving truck so nothing rubs or hits it, and avoid eating in bed as well so there’s no chance of anything spilling on your frame. If you’re still stumped on how to design your bedroom or what set looks best, we have a blog on How to Style Your Bedroom in 3 Ways where you can get that extra dose of inspiration you’ve been looking for!

What Type of Bedroom Set is Best for My Style?
If your priority is storage, be sure to look at master bedroom sets that include bed storage with drawers or a footboard with shelves. For something timeless and beautiful, opt for beds with detailed wood grain. Tufted styles on the headboard allow for a more feminine and elegant presentation, while sleigh beds have an impressive silhouette. Flank a pair of nightstands and chic table lamps on top, and you’ll never go wrong with a bedroom set.


In modern computational chemistry, quantum chemical calculations are performed using a finite set of basis functions. When the finite basis is expanded towards an (infinite) complete set of functions, calculations using such a basis set are said to approach the complete basis set (CBS) limit. In this context, basis function and atomic orbital are sometimes used interchangeably, although the basis functions are usually not true atomic orbitals.

Within the basis set, the wavefunction is represented as a vector, the components of which correspond to coefficients of the basis functions in the linear expansion. In such a basis, one-electron operators correspond to matrices (a.k.a. rank two tensors), whereas two-electron operators are rank four tensors.

When molecular calculations are performed, it is common to use a basis composed of atomic orbitals, centered at each nucleus within the molecule (linear combination of atomic orbitals ansatz). The physically best motivated basis set are Slater-type orbitals (STOs), which are solutions to the Schrödinger equation of hydrogen-like atoms, and decay exponentially far away from the nucleus. It can be shown that the molecular orbitals of Hartree-Fock and density-functional theory also exhibit exponential decay. Furthermore, S-type STOs also satisfy Kato's cusp condition at the nucleus, meaning that they are able to accurately describe electron density near the nucleus. However, hydrogen-like atoms lack many-electron interactions, thus the orbitals do not accurately describe electron state correlations.

Unfortunately, calculating integrals with STOs is computationally difficult and it was later realized by Frank Boys that STOs could be approximated as linear combinations of Gaussian-type orbitals (GTOs) instead. Because the product of two GTOs can be written as a linear combination of GTOs, integrals with Gaussian basis functions can be written in closed form, which leads to huge computational savings (see John Pople).

Dozens of Gaussian-type orbital basis sets have been published in the literature. [2] Basis sets typically come in hierarchies of increasing size, giving a controlled way to obtain more accurate solutions, however at a higher cost.

The smallest basis sets are called minimal basis sets. A minimal basis set is one in which, on each atom in the molecule, a single basis function is used for each orbital in a Hartree–Fock calculation on the free atom. For atoms such as lithium, basis functions of p type are also added to the basis functions that correspond to the 1s and 2s orbitals of the free atom, because lithium also has a 1s2p bound state. For example, each atom in the second period of the periodic system (Li - Ne) would have a basis set of five functions (two s functions and three p functions).

A minimal basis set may already be exact for the gas-phase atom at the self-consistent field level of theory. In the next level, additional functions are added to describe polarization of the electron density of the atom in molecules. These are called polarization functions. For example, while the minimal basis set for hydrogen is one function approximating the 1s atomic orbital, a simple polarized basis set typically has two s- and one p-function (which consists of three basis functions: px, py and pz). This adds flexibility to the basis set, effectively allowing molecular orbitals involving the hydrogen atom to be more asymmetric about the hydrogen nucleus. This is very important for modeling chemical bonding, because the bonds are often polarized. Similarly, d-type functions can be added to a basis set with valence p orbitals, and f-functions to a basis set with d-type orbitals, and so on.

Another common addition to basis sets is the addition of diffuse functions. These are extended Gaussian basis functions with a small exponent, which give flexibility to the "tail" portion of the atomic orbitals, far away from the nucleus. Diffuse basis functions are important for describing anions or dipole moments, but they can also be important for accurate modeling of intra- and intermolecular bonding.

The most common minimal basis set is STO-nG, where n is an integer. The STO-nG basis sets are derived from a minimal Slater-type orbital basis set, with n representing the number of Gaussian primitive functions used to represent each Slater-type orbital. Minimal basis sets typically give rough results that are insufficient for research-quality publication, but are much cheaper than their larger counterparts. Commonly used minimal basis sets of this type are:

There are several other minimum basis sets that have been used such as the MidiX basis sets.

During most molecular bonding, it is the valence electrons which principally take part in the bonding. In recognition of this fact, it is common to represent valence orbitals by more than one basis function (each of which can in turn be composed of a fixed linear combination of primitive Gaussian functions). Basis sets in which there are multiple basis functions corresponding to each valence atomic orbital are called valence double, triple, quadruple-zeta, and so on, basis sets (zeta, ζ, was commonly used to represent the exponent of an STO basis function [4] ). Since the different orbitals of the split have different spatial extents, the combination allows the electron density to adjust its spatial extent appropriate to the particular molecular environment. In contrast, minimal basis sets lack the flexibility to adjust to different molecular environments.

Pople basis sets Edit

The notation for the split-valence basis sets arising from the group of John Pople is typically X-YZg. [5] In this case, X represents the number of primitive Gaussians comprising each core atomic orbital basis function. The Y and Z indicate that the valence orbitals are composed of two basis functions each, the first one composed of a linear combination of Y primitive Gaussian functions, the other composed of a linear combination of Z primitive Gaussian functions. In this case, the presence of two numbers after the hyphens implies that this basis set is a split-valence double-zeta basis set. Split-valence triple- and quadruple-zeta basis sets are also used, denoted as X-YZWg, X-YZWVg, etc. Here is a list of commonly used split-valence basis sets of this type:

  • 3-21G
  • 3-21G* - Polarization functions on heavy atoms
  • 3-21G** - Polarization functions on heavy atoms and hydrogen
  • 3-21+G - Diffuse functions on heavy atoms
  • 3-21++G - Diffuse functions on heavy atoms and hydrogen
  • 3-21+G* - Polarization and diffuse functions on heavy atoms
  • 3-21+G** - Polarization functions on heavy atoms and hydrogen, as well as diffuse functions on heavy atoms
  • 4-21G
  • 4-31G
  • 6-21G
  • 6-31G
  • 6-31G*
  • 6-31+G*
  • 6-31G(3df, 3pd)
  • 6-311G
  • 6-311G*
  • 6-311+G*

The 6-31G* basis set (defined for the atoms H through Zn) is a valence double-zeta polarized basis set that adds to the 6-31G set five d-type Cartesian-Gaussian polarization functions on each of the atoms Li through Ca and ten f-type Cartesian Gaussian polarization functions on each of the atoms Sc through Zn.

As compared to Pople basis sets, correlation-consistent or polarization-consistent basis sets are more appropriate for correlated wave function calculations. [6] For Hartree-Fock or density functional theory, however, Pople basis sets are more efficient (per unit basis function) as compared to other alternatives, provided that the electronic structure program can take advantage of combined sp shells.

Some of the most widely used basis sets are those developed by Dunning and coworkers, [7] since they are designed for converging Post-Hartree–Fock calculations systematically to the complete basis set limit using empirical extrapolation techniques.

For first- and second-row atoms, the basis sets are cc-pVNZ where N=D,T,Q,5,6. (D=double, T=triples, etc.). The 'cc-p', stands for 'correlation-consistent polarized' and the 'V' indicates they are valence-only basis sets. They include successively larger shells of polarization (correlating) functions (d, f, g, etc.). More recently these 'correlation-consistent polarized' basis sets have become widely used and are the current state of the art for correlated or post-Hartree–Fock calculations. Examples of these are:

  • cc-pVDZ - Double-zeta
  • cc-pVTZ - Triple-zeta
  • cc-pVQZ - Quadruple-zeta
  • cc-pV5Z - Quintuple-zeta, etc.
  • aug-cc-pVDZ, etc. - Augmented versions of the preceding basis sets with added diffuse functions.
  • cc-pCVDZ - Double-zeta with core correlation

For period-3 atoms (Al-Ar), additional functions have turned out to be necessary these are the cc-pV(N+d)Z basis sets. Even larger atoms may employ pseudopotential basis sets, cc-pVNZ-PP, or relativistic-contracted Douglas-Kroll basis sets, cc-pVNZ-DK.

While the usual Dunning basis sets are for valence-only calculations, the sets can be augmented with further functions that describe core electron correlation. These core-valence sets (cc-pCVXZ) can be used to approach the exact solution to the all-electron problem, and they are necessary for accurate geometric and nuclear property calculations.

Weighted core-valence sets (cc-pwCVXZ) have also been recently suggested. The weighted sets aim to capture core-valence correlation, while neglecting most of core-core correlation, in order to yield accurate geometries with smaller cost than the cc-pCVXZ sets.

Diffuse functions can also be added for describing anions and long-range interactions such as Van der Waals forces, or to perform electronic excited-state calculations, electric field property calculations. A recipe for constructing additional augmented functions exists as many as five augmented functions have been used in second hyperpolarizability calculations in the literature. Because of the rigorous construction of these basis sets, extrapolation can be done for almost any energetic property. However, care must be taken when extrapolating energy differences as the individual energy components converge at different rates: the Hartree-Fock energy converges exponentially, whereas the correlation energy converges only polynomially.

H-He Li-Ne Na-Ar
cc-pVDZ [2s1p] → 5 func. [3s2p1d] → 14 func. [4s3p1d] → 18 func.
cc-pVTZ [3s2p1d] → 14 func. [4s3p2d1f] → 30 func. [5s4p2d1f] → 34 func.
cc-pVQZ [4s3p2d1f] → 30 func. [5s4p3d2f1g] → 55 func. [6s5p3d2f1g] → 59 func.
aug-cc-pVDZ [3s2p] → 9 func. [4s3p2d] → 23 func. [5s4p2d] → 27 func.
aug-cc-pVTZ [4s3p2d] → 23 func. [5s4p3d2f] → 46 func. [6s5p3d2f] → 50 func.
aug-cc-pVQZ [5s4p3d2f] → 46 func. [6s5p4d3f2g] → 80 func. [7s6p4d3f2g] → 84 func.

To understand how to get the number of functions take the cc-pVDZ basis set for H: There are two s (L = 0) orbitals and one p (L = 1) orbital that has 3 components along the z-axis (mL = -1,0,1) corresponding to px, py and pz. Thus, five spatial orbitals in total. Note that each orbital can hold two electrons of opposite spin.

For example, Ar [1s, 2s, 2p, 3s, 3p] has 3 s orbitals (L=0) and 2 sets of p orbitals (L=1). Using cc-pVDZ, orbitals are [1s, 2s, 2p, 3s, 3s, 3p, 3p, 3d'] (where ' represents the added in polarisation orbitals), with 4 s orbitals (4 basis functions), 3 sets of p orbitals (3 × 3 = 9 basis functions), and 1 set of d orbitals (5 basis functions). Adding up the basis functions gives a total of 18 functions for Ar with the cc-pVDZ basis-set.

Density-functional theory has recently become widely used in computational chemistry. However, the correlation-consistent basis sets described above are suboptimal for density-functional theory, because the correlation-consistent sets have been designed for Post-Hartree–Fock, while density-functional theory exhibits much more rapid basis set convergence than wave function methods.

Adopting a similar methodology to the correlation-consistent series, Frank Jensen introduced polarization-consistent (pc-n) basis sets as a way to quickly converge density functional theory calculations to the complete basis set limit. [8] Like the Dunning sets, the pc-n sets can be combined with basis set extrapolation techniques to obtain CBS values.

The pc-n sets can be augmented with diffuse functions to obtain augpc-n sets.

Some of the various valence adaptations of Karlsruhe basis sets are

  • def2-SV(P) - Split valence with polarization functions on heavy atoms (not hydrogen)
  • def2-SVP - Split valence polarization
  • def2-SVPD - Split valence polarization with diffuse functions
  • def2-TZVP - Valence triple-zeta polarization
  • def2-TZVPD - Valence triple-zeta polarization with diffuse functions
  • def2-TZVPP - Valence triple-zeta with two sets of polarization functions
  • def2-TZVPPD - Valence triple-zeta with two sets of polarization functions and a set of diffuse functions
  • def2-QZVP - Valence quadruple-zeta polarization
  • def2-QZVPD - Valence quadruple-zeta polarization with diffuse functions
  • def2-QZVPP - Valence quadruple-zeta with two sets of polarization functions
  • def2-QZVPPD - Valence quadruple-zeta with two sets of polarization functions and a set of diffuse functions

Gaussian-type orbital basis sets are typically optimized to reproduce the lowest possible energy for the systems used to train the basis set. However, the convergence of the energy does not imply convergence of other properties, such as nuclear magnetic shieldings, the dipole moment, or the electron momentum density, which probe different aspects of the electronic wave function.

Manninen and Vaara have proposed completeness-optimized basis sets, [9] where the exponents are obtained by maximization of the one-electron completeness profile [10] instead of minimization of the energy. Completeness-optimized basis sets are a way to easily approach the complete basis set limit of any property at any level of theory, and the procedure is simple to automatize. [11]

Completeness-optimized basis sets are tailored to a specific property. This way, the flexibility of the basis set can be focused on the computational demands of the chosen property, typically yielding much faster convergence to the complete basis set limit than is achievable with energy-optimized basis sets.

In 1974 Bardo and Ruedenberg [12] proposed a simple scheme to generate the exponents of a basis set that spans the Hilbert space evenly [13] by following a geometric progression of the form:

In addition to localized basis sets, plane-wave basis sets can also be used in quantum-chemical simulations. Typically, the choice of the plane wave basis set is based on a cutoff energy. The plane waves in the simulation cell that fit below the energy criterion are then included in the calculation. These basis sets are popular in calculations involving three-dimensional periodic boundary conditions.

The main advantage of a plane-wave basis is that it is guaranteed to converge in a smooth, monotonic manner to the target wavefunction. In contrast, when localized basis sets are used, monotonic convergence to the basis set limit may be difficult due to problems with over-completeness: in a large basis set, functions on different atoms start to look alike, and many eigenvalues of the overlap matrix approach zero.

In addition, certain integrals and operations are much easier to program and carry out with plane-wave basis functions than with their localized counterparts. For example, the kinetic energy operator is diagonal in the reciprocal space. Integrals over real-space operators can be efficiently carried out using fast Fourier transforms. The properties of the Fourier Transform allow a vector representing the gradient of the total energy with respect to the plane-wave coefficients to be calculated with a computational effort that scales as NPW*ln(NPW) where NPW is the number of plane-waves. When this property is combined with separable pseudopotentials of the Kleinman-Bylander type and pre-conditioned conjugate gradient solution techniques, the dynamic simulation of periodic problems containing hundreds of atoms becomes possible.

In practice, plane-wave basis sets are often used in combination with an 'effective core potential' or pseudopotential, so that the plane waves are only used to describe the valence charge density. This is because core electrons tend to be concentrated very close to the atomic nuclei, resulting in large wavefunction and density gradients near the nuclei which are not easily described by a plane-wave basis set unless a very high energy cutoff, and therefore small wavelength, is used. This combined method of a plane-wave basis set with a core pseudopotential is often abbreviated as a PSPW calculation.

Furthermore, as all functions in the basis are mutually orthogonal and are not associated with any particular atom, plane-wave basis sets do not exhibit basis-set superposition error. However, the plane-wave basis set is dependent on the size of the simulation cell, complicating cell size optimization.

Due to the assumption of periodic boundary conditions, plane-wave basis sets are less well suited to gas-phase calculations than localized basis sets. Large regions of vacuum need to be added on all sides of the gas-phase molecule in order to avoid interactions with the molecule and its periodic copies. However, the plane waves use a similar accuracy to describe the vacuum region as the region where the molecule is, meaning that obtaining the truly noninteracting limit may be computationally costly.

Real-space approaches offer powerful methods to solve electronic structure problems thanks to their controllable accuracy. Real-space basis sets can be thought to arise from the theory of interpolation, as the central idea is to represent the (unknown) orbitals in terms of some set of interpolation functions.

Various methods have been proposed for constructing the solution in real space, including finite elements, basis splines, Lagrange sinc-functions, and wavelets. [1] Finite difference algorithms are also often included in this category, even though precisely speaking, they do not form a proper basis set and are not variational unlike e.g. finite element methods. [1]

A common feature of all real-space methods is that the accuracy of the numerical basis set is improvable, so that the complete basis set limit can be reached in a systematical manner. Moreover, in the case of wavelets and finite elements, it is easy to use different levels of accuracy in different parts of the system, so that more points are used close to the nuclei where the wave function undergoes rapid changes and where most of the total energies lie, whereas a coarser representation is sufficient far away from nuclei this feature is extremely important as it can be used to make all-electron calculations tractable.

  1. ^ abc Lehtola, Susi (2019). "A review on non-relativistic fully numerical electronic structure calculations on atoms and diatomic molecules". Int. J. Quantum Chem. 119: e25968. doi: 10.1002/qua.25968 .
  2. ^
  3. Jensen, Frank (2013). "Atomic orbital basis sets". WIREs Comput. Mol. Sci. 3 (3): 273–295. doi:10.1002/wcms.1123.
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  5. Errol G. Lewars (2003-01-01). Computational Chemistry: Introduction to the Theory and Applications of Molecular and Quantum Mechanics (1st ed.). Springer. ISBN978-1402072857 .
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  7. Davidson, Ernest Feller, David (1986). "Basis set selection for molecular calculations". Chem. Rev. 86 (4): 681–696. doi:10.1021/cr00074a002.
  8. ^
  9. Ditchfield, R Hehre, W.J Pople, J. A. (1971). "Self-Consistent Molecular-Orbital Methods. IX. An Extended Gaussian-Type Basis for Molecular-Orbital Studies of Organic Molecules". J. Chem. Phys. 54 (2): 724–728. Bibcode:1971JChPh..54..724D. doi:10.1063/1.1674902.
  10. ^
  11. Moran, Damian Simmonett, Andrew C. Leach, Franklin E. III Allen, Wesley D. Schleyer, Paul v. R. Schaefer, Henry F. (2006). "Popular theoretical methods predict benzene and arenes to be nonplanar". J. Am. Chem. Soc. 128 (29): 9342–9343. doi:10.1021/ja0630285. PMID16848464.
  12. ^
  13. Dunning, Thomas H. (1989). "Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen". J. Chem. Phys. 90 (2): 1007–1023. Bibcode:1989JChPh..90.1007D. doi:10.1063/1.456153.
  14. ^
  15. Jensen, Frank (2001). "Polarization consistent basis sets: Principles". J. Chem. Phys. 115 (20): 9113–9125. Bibcode:2001JChPh.115.9113J. doi:10.1063/1.1413524.
  16. ^
  17. Manninen, Pekka Vaara, Juha (2006). "Systematic Gaussian basis-set limit using completeness-optimized primitive sets. A case for magnetic properties". J. Comput. Chem. 27 (4): 434–445. doi:10.1002/jcc.20358. PMID16419020.
  18. ^
  19. Chong, Delano P. (1995). "Completeness profiles of one-electron basis sets". Can. J. Chem. 73 (1): 79–83. doi:10.1139/v95-011.
  20. ^
  21. Lehtola, Susi (2015). "Automatic algorithms for completeness-optimization of Gaussian basis sets". J. Comput. Chem. 36 (5): 335–347. doi:10.1002/jcc.23802. PMID25487276.
  22. ^
  23. Bardo, Richard D. Ruedenberg, Klaus (February 1974). "Even‐tempered atomic orbitals. VI. Optimal orbital exponents and optimal contractions of Gaussian primitives for hydrogen, carbon, and oxygen in molecules". The Journal of Chemical Physics. 60 (3): 918–931. Bibcode:1974JChPh..60..918B. doi:10.1063/1.1681168. ISSN0021-9606.
  24. ^ ab
  25. Cherkes, Ira Klaiman, Shachar Moiseyev, Nimrod (2009-11-05). "Spanning the Hilbert space with an even tempered Gaussian basis set". International Journal of Quantum Chemistry. 109 (13): 2996–3002. Bibcode:2009IJQC..109.2996C. doi:10.1002/qua.22090.
  26. ^
  27. Nakai, Hiromi (2002). "Simultaneous determination of nuclear and electronic wave functions without Born-Oppenheimer approximation: Ab initio NO+MO/HF theory". International Journal of Quantum Chemistry. 86 (6): 511–517. doi:10.1002/qua.1106. ISSN0020-7608.
  28. ^
  29. Moncada, Félix Cruz, Daniel Reyes, Andrés (June 2012). "Muonic alchemy: Transmuting elements with the inclusion of negative muons". Chemical Physics Letters. 539–540: 209–213. Bibcode:2012CPL. 539..209M. doi:10.1016/j.cplett.2012.04.062.
  30. ^
  31. Reyes, Andrés Moncada, Félix Charry, Jorge (2019-01-15). "The any particle molecular orbital approach: A short review of the theory and applications". International Journal of Quantum Chemistry. 119 (2): e25705. doi: 10.1002/qua.25705 . ISSN0020-7608.
  32. ^
  33. Lehtola, Susi (2019). "Fully numerical Hartree–Fock and density functional calculations. I. Atoms". Int. J. Quantum Chem. 119: e25945. doi:10.1002/qua.25945. hdl: 10138/311128 .

All the many basis sets discussed here along with others are discussed in the references below which themselves give references to the original journal articles:

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Microsoft shares nightmare tale: 6 sets of hackers on a customer's network

Microsoft reveals its first report on incident response work carried out by its Detection and Response Team (DART).

By Liam Tung | March 10, 2020 -- 13:00 GMT (06:00 PDT) | Topic: Security

Microsoft's first report from its Detection and Response Team (DART), which helps customers in deep cyber trouble, details the case of a large customer with six threat actors simultaneously on its network, including one state-sponsored hacker group that had been stealing data and email for 243 days.

The company announced DART in March 2019 as part of the $1bn-a-year push into enterprise cybersecurity announced by CEO Satya Nadella in 2017.


Without revealing any customer names, Microsoft intends to publish regular updates about DART's activities, to illustrate how hackers are operating.

Its first report details an advanced persistent threat (APT) attacker that stole administrator credentials to penetrate the target's network and steal sensitive data and emails.

Notably, the customer was not using multi-factor authentication (MFA), which could have prevented the breach. Microsoft revealed last week that 99.9% of compromised accounts didn't use MFA, and only 11% of enterprise accounts use MFA.

DART was brought in after the customer failed to kick one APT attacker off its network after 243 days, despite having engaged an incident response vendor seven months earlier. The attacker was ejected on the day Microsoft's team arrived. It also discovered five other threat groups were inside the network.

In this case, the main attacker used a password-spraying attack to grab the customer's Office 365 admin credentials and from there searched mailboxes to find more credentials shared among employees in emails. DART found the attacker was looking for intellectual property in certain markets.

The attacker even used the customer's e-discovery and compliance tools to automate the search for relevant emails.

According to Microsoft, the company in the first month of the attack tried to handle the compromised Office 365 account itself, then brought in an incident-response vendor to lead what turned out to be a lengthy investigation.

"This investigation lasted more than seven months and revealed a possible compromise of sensitive information – pertaining to the victim and the victim's customers – stored in Office 365 mailboxes. 243 days after the initial compromise, DART was then brought in to work alongside the incident-response vendor and the company's in-house teams," Microsoft says.

"DART quickly identified targeted mailbox searches and compromised accounts, as well as attacker command-and-control channels. DART also identified five additional, distinct attacker campaigns persisting in the environment that were unrelated to the initial incident. They discovered these attackers had entered the environment even earlier to establish access channels (ie, back doors) for later use as needed."

Microsoft outlines five basic steps that organizations can use to minimize their exposure to APT attackers, including enabling MFA, removing legacy authentication, adequately training first responders, properly logging events with a security, information and event management product, and recognizing that attackers do use legitimate administrative and security tools to probe targets.

The post offers the same message it gave to customers who are victims of major ransomware groups last week: customers should enable available security tools and focus on logging security events.

Microsoft covered the work of the operators of REvil, Samas or SamSam, Doppelpaymer, Bitpaymer, and Ryuk ransomware. It detailed how attackers disable security software and noted that some customers even disable security software to improve performance, allowing cybercriminals to roam networks for months unfettered.